# Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

aa r X i v : . [ m a t h . R T ] F e b INFINITE-DIMENSIONAL GROUPS OVER FINITE FIELDS ANDHALL-LITTLEWOOD SYMMETRIC FUNCTIONS

CESAR CUENCA AND GRIGORI OLSHANSKI

Abstract.

The groups mentioned in the title are certain matrix groups of inﬁnite size withelements in a ﬁnite ﬁeld. They are built from ﬁnite classical groups and at the same timethey are similar to reductive p -adic Lie groups. In the present paper, we initiate the study ofinvariant measures for the coadjoint action of these inﬁnite-dimensional groups. Of specialinterest for us are ergodic invariant measures, which are a substitute of orbital measures.We examine ﬁrst the group GLB , a topological completion of the inductive limit grouplim −→ GL ( n, q ), where q is the cardinality of the ﬁnite ﬁeld. As was shown by Gorin, Kerov,and Vershik [11], GLB has a rich family of traceable factor representations (a reasonablesubstitute of irreducible representations for a non-type I group). These representations ad-mit a complete classiﬁcation, which is achieved in terms of harmonic functions on the Younggraph Y , as in the classical Thoma’s theorem referring to traceable factor representationsof the inﬁnite symmetric group.We show that there exists a parallel theory for ergodic coadjoint-invariant measures,which is linked with harmonic functions on the “HL-deformed Young graph” Y HL ( t ). Herethe deformation means that the edges of Y are endowed with certain formal multiplicitiescoming from the simplest version of Pieri rule (multiplication by the ﬁrst power sum p )for the Hall–Littlewood (HL) symmetric functions, and t is the HL parameter specialized to q − .This fact serves as a prelude to our main results, which concern topological completionsof two inductive limit groups built from ﬁnite unitary groups, of even and odd dimension.We show that in this case, coadjoint-invariant measures are linked to some new branchinggraphs. The latter are still related to the HL functions, but the principal novelty is that nowthe formal edge multiplicities come from the structure constants of the multiplication by p ,not p . Another novel feature is that the HL parameter t turns out to be negative and equalto − q − . This is a new version of Ennola’s duality [7, 8], which is the principle that therepresentations of the unitary and general linear groups over a ﬁnite ﬁeld are connected bya sign ﬂip: q ↔ − q . As a ﬁrst application of our results, we ﬁnd several families of unitarilyinvariant measures, including analogues of the Plancherel measure. Contents

1. Introduction 31.1. Setting of the problem 31.2. Motivation 41.3. The results 51.4. Acknowledgements 7

Part 1. GL ( ∞ , q ) -invariant measures

72. The ﬁnite general linear group: preliminaries 72.1. The group GL ( n, q ) and the Lie algebra gl ( n, q ) 72.2. Partitions 7 gl ( n, q ) 82.4. General conjugacy classes in gl ( n, q ) 83. Invariant measures on L ( q ): generalities 83.1. The space L ( q ). The groups GL ( ∞ , q ) and GLB L ( q ) 93.3. Invariant measures on L ( q ) 103.4. All nontrivial invariant Radon measures on L ( q ) are inﬁnite 113.5. Invariant measures on pronilpotent matrices 114. Description of P GL ( ∞ ,q )0 Y HL ( t ) and its boundary 134.3. The branching graph Γ GL ( ∞ ,q ) P GL ( ∞ ,q ) L ( q ) 195.2. The structure of P GL ( ∞ ,q ) Part 2. U (2 ∞ , q ) - and U (2 ∞ + 1 , q ) -invariant measures u ( N, q ) 236.3. General conjugacy classes in u ( N, q ) 247. Invariant measures on skew-Hermitian matrices 257.1. The group U (2 ∞ , q ) and the space L E ( q ) 257.2. The group U (2 ∞ + 1 , q ) and the space L O ( q ) 277.3. Invariant Radon measures 277.4. Relationship among P U (2 ∞ ,q ) , P U (2 ∞ +1 ,q ) , P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 P U (2 ∞ ,q )0 , P U (2 ∞ +1 ,q )0 and Ennola’s duality 298.1. The branching graphs Γ U (2 ∞ ,q ) and Γ U (2 ∞ +1 ,q ) Part 3. Appendices Introduction

Setting of the problem.

Fix a ﬁnite ﬁeld F q with q elements and let { G ( n ) } be anyof the classical series of ﬁnite groups of Lie type over F q . The basic example is that ofgeneral linear groups G ( n ) = GL ( n, q ) := GL ( n, F q ), but one can also consider the unitary,orthogonal or symplectic groups over a ﬁnite ﬁeld. Such groups G ( n ) form a nested chain,so one can form the inductive limit group G := lim −→ G ( n ). The “inﬁnite-dimensional groupsover F q ” mentioned in the title are certain topological completions G ⊃ G of such inductivelimit groups.In the case of general linear groups, the deﬁnition of the topological group G is given inthe work [11] by Gorin, Kerov, and Vershik. This group, denoted in [11] as GLB , is formedby the inﬁnite size matrices g = [ g ij ] ∞ i,j =1 over F q which have ﬁnitely many nonzero entriesbelow the diagonal and are invertible. It is a locally compact, separable, totally disconnectedtopological group with respect to a natural nondiscrete topology. That topology is uniquelydeﬁned by the condition that the subgroup B ⊂ GLB of upper triangular matrices (which isa proﬁnite group) is an open compact subgroup.For the other classical series, the corresponding groups G are deﬁned in a similar way.Although these groups are not Lie groups in the proper sense, one can deﬁne for themnatural analogues of a Lie algebra, adjoint and coadjoint action.For example, consider the case G = GLB . Then the corresponding Lie algebra ¯ g is formedby the inﬁnite size matrices X = [ X ij ] ∞ i,j =1 over F q with ﬁnitely many nonzero entries belowthe diagonal. It is a locally compact vector space over F q : the topology is deﬁned so thatthe subspace of upper triangular matrices (which is a proﬁnite group under addition) is anopen compact subgroup. Next, let ¯ g ∗ be the set of inﬁnite size matrices Y = [ Y ij ] ∞ i,j =1 witha modiﬁed ﬁniteness condition: Y must have ﬁnitely many nonzero elements both belowand on the diagonal. The topology on ¯ g ∗ is deﬁned in a similar way, but now the subspaceof strictly upper triangular matrices is an open compact subgroup. The topological vectorspaces ¯ g and ¯ g ∗ are dual to each over: the duality between them is given by the bilinear map¯ g × ¯ g ∗ → F , ( X, Y ) Tr( XY ) . They are also dual to each other as commutative locally compact groups. Finally, note thatthe group G = GLB acts on ¯ g and on ¯ g ∗ by conjugation, and these are the adjoint andcoadjoint actions in question. Again, this deﬁnition can be easily extended to the otherseries.We raise the following problem. Problem 1.1.

Let G and ¯ g ∗ be as above. Study the invariant Radon measures for thecoadjoint action of G on ¯ g ∗ . In particular, describe the ergodic invariant Radon measures.Recall that a Radon measure on a locally compact space is a possibly inﬁnite measuretaking ﬁnite values on compact subsets. In our situation, all invariant measures turn out tobe inﬁnite, with a trivial exception. Note also that in the formulation of the problem, thetopological group G can be replaced by its countable subgroup G ⊂ G (the inductive limitsubgroup lim −→ G ( n )) — the set of invariant measures will be the same. Remark 1.2. (a) With the only exception of the delta measure at { } , the G -invariantRadon measures on ¯ g ∗ are inﬁnite measures. Despite this, there are some points of contactsbetween our theory and Fulman’s probabilistic theory of random matrices over ﬁnite ﬁelds CESAR CUENCA AND GRIGORI OLSHANSKI (see his survey paper [9] and references therein). In particular, in both cases, Hall–Littlewoodsymmetric functions play a fundamental role.(b) In the case of a compact group action on a locally compact space, ergodic invariantmeasures are the same as orbital measures (that is, invariant measures concentrated on theorbits). Our situation is very diﬀerent. Namely, there are plenty of orbital measures, relatedto arbitrary G -orbits in ¯ g ∗ ; they are all ergodic, but typically fail to be Radon measures.Thus, our ergodic invariant Radon measures are typically not orbital measures.(c) The Radon condition is substantially used in our paper. Note that if P is a Radonmeasure on ¯ g ∗ , then one can deﬁne its Fourier transform b P , which is a distribution on ¯ g .1.2. Motivation.

Problem 1.1 is prompted by the work of Gorin-Kerov-Vershik [11] onunitary representations of the group

GLB (see also the announcements [22], [23]). Since

GLB is not a type I group, classifying its irreducible representations is a wild problem.However, there is a reasonable substitute of irreducible representations — these are thetraceable factor representations. The latter are deﬁned by the indecomposable traces onthe subalgebra A ( GLB ) ⊂ L ( GLB ) formed by the locally constant, compactly supportedfunctions on

GLB .One of the main results of [11] is a solution of the classiﬁcation problem for the indecom-posable traces on A ( GLB ). Here is its brief description:(1) the whole set of indecomposable traces can be partitioned into countably many families;(2) each family is in a natural one-to-one correspondence with the set H + ( Y ) of nonneg-ative harmonic functions on the Young graph Y .By deﬁnition, the vertex set of the graph Y (denoted by the same symbol Y ) consists ofYoung diagrams, and the edges are formed by pairs µ ⊂ λ of Young diagrams which diﬀer bya single box (then we write µ ր λ ). The elements of H + ( Y ) are the functions ϕ : Y → R ≥ subject to the harmonicity condition ϕ ( µ ) = X λ ∈ Y : µ ր λ ϕ ( λ ) , µ ∈ Y . (1.1)From this equation, it is seen that H + ( Y ) is a convex cone, and a general theorem assertsthat it is isomorphic to the cone of ﬁnite measures on a certain space that is associated withthe graph and called its boundary . The explicit description of the boundary of Y is known:it is an inﬁnite-dimensional compact space (the Thoma simplex). This is a classical result(equivalent to Thoma’s theorem about ﬁnite factor representations of the inﬁnite symmetricgroup). It follows that the indecomposable functions ϕ ∈ H + ( Y ) (that is, the elements ofthe extreme rays of the cone) correspond to the points of the Thoma simplex, which inturn depend on countably many continuous parameters. This ﬁnally leads to an explicitparametrization of the indecomposable traces on A ( GLB ).The work [11] raises a number of open problems, one of which (probably, the most evidentone) is the study of the traces on A ( G ) for other classical series.Problem 1.1 is in fact a variation of that problem. To see this, we remark that if elementsof A ( G ) are treated as test functions, then the traces are precisely the positive deﬁnitedistributions on G , invariant under the action of G by inner automorphisms. On the otherhand, the Fourier transforms of the coadjoint-invariant Radon measures on ¯ g ∗ are preciselythe positive deﬁnite distributions on the vector space ¯ g , invariant under the adjoint actionof G . In this interpretation, an analogy between the two kinds of objects becomes apparent(in particular, ergodic invariant measures can be treated as a counterpart of indecomposable traces). This is a manifestation of a parallelism between problems referring to Lie groups(linked to characters) and to Lie algebras (linked to conjugacy classes), which arises in agreat variety of situations.It seems that the study of G -invariant measures on ¯ g ∗ can be both easier and more diﬃcultthan the study of traces on A ( G ). On the one hand, in contrast to irreducible charactersof ﬁnite classical groups, the parametrization of conjugacy classes in all ﬁnite-dimensionalclassical Lie algebras over a ﬁnite ﬁeld is achieved by tools of linear algebra (Wall [24],Burgoyne–Cushman [6]). On the other hand, as can be seen from the comparison of (1.1)and (1.3), in the case of invariant measures, some of the combinatorial structures that arisemay be more involved.1.3. The results.

The paper is divided into three parts. The material of Part 1 mainlyserves us as a guiding example, Part 2 contains the main results, and Part 3 consists of twoappendices.1.3.1. In Part 1 (sections 2–5) we examine the case of the general linear groups, which issimpler than that of other classical groups. Thus, in Part 1, G is the group GLB . For thespace ¯ g ∗ , we use the alternate notation L ( q ). Let P denote the convex cone of invariantRadon measures on L ( q ). We show that P splits into a direct product of countably manyconvex cones: P = Y σ ∈ Σ P σ (Σ is an index set). (1.2)This decomposition comes from certain partition of L ( q ) into a disjoint union of subsets whichare invariant and clopen (i.e. both closed and open). Moreover, all the cones P σ are pairwiseisomorphic, so we focus on the description of one of them, a distinguished cone denoted by P : it is formed by the invariant measures supported on the subset Nil( L ( q )) ⊂ L ( q ) of pronilpotent matrices .We show that P is isomorphic to the cone of nonnegative harmonic functions on a certain branching graph , denoted by Y HL ( q − ). For a general parameter t ranging over (0 , Y HL ( t ) is a t -deformation of the Young graph Y in the following sense.The two graphs have common vertices and edges, but in Y HL ( t ), the edges are endowedwith certain formal multiplicities : these are the coeﬃcients ψ λ/µ ( t ) in the simplest Pieri rule(multiplication by the ﬁrst power sum p ) for the Hall–Littlewood (HL) symmetric functionswith parameter t . In accordance to this, the harmonic functions in this setting satisfy adeformed version of equation (1.1), which is obtained by inserting the coeﬃcients ψ λ/µ ( t ) onthe right-hand side; it takes the form ϕ ( µ ) = X λ ∈ Y : µ ր λ ψ λ/µ ( t ) ϕ ( λ ) , µ ∈ Y . (1.3)Thus, the picture looks quite similar to what we described above: the problem is reduced toﬁnding the boundary of a branching graph — the HL-deformed graph Y HL ( t ) with t = q − .Now we can apply results of Kerov [12] and Matveev [17] to complete the classiﬁcation.Namely, Kerov suggested an ingenious way to construct harmonic functions correspondingto boundary points (in fact, in a more general setting; see Borodin–Corwin [3, Sect. 2.2.1] fordetails), and Matveev recently managed to prove that Kerov’s construction gives exactly allelements of the boundary. In this way we are able to obtain a complete answer to Problem1.1 in the case of G = GLB . CESAR CUENCA AND GRIGORI OLSHANSKI

Note that the central result of part 1, the isomorphism between the cones P and H + ( Y HL ( q − )),is essentially equivalent to Theorem 4.6 in [11] concerning the so-called central measures onthe subgroup U ⊂ GLB of upper unitriangular matrices. However, we present the materialfrom another perspective and also give a number of other results.1.3.2. In Part 2 (sections 6–8) we examine the case of unitary groups. The ﬁeld F q hasa unique quadratic extension F q , which makes it possible to deﬁne sesquilinear Hermitianforms. In each dimension N , there is only one, within equivalence, nondegenerate sesquilinearHermitian form, and we denote by U ( N, q ) the corresponding unitary group. It can berealized as a subgroup of GL ( N, q ) in various ways, depending on the choice of the matrixof the form. For building a GLB -like topological completion G of an inductive limit group G = lim −→ G ( n ), we need embeddings G ( n ) → G ( n + 1) which are consistent with Borelsubgroups. To satisfy this condition, we have to consider separately two twin series { G ( n ) } and two direct limit unitary groups, which we call informally even and odd (this term refersto the parity of N ): U (2 ∞ , q ) := lim −→ U (2 n, q ) , U (2 ∞ + 1 , q ) := lim −→ U (2 n + 1 , q ) . In both series, the matrices of sesquilinear forms have 1’s along the secondary diagonal and0’s elsewhere — this leads to the required embeddings of groups and makes it possible forus to deﬁne the two desired topological completions.Next, we deﬁne, in a natural way, the two corresponding Lie algebras, their dual spaces,and the coadjoint actions. Let P E and P O be the cones formed by the coadjoint-invariantRadon measures (here E and O are abbreviations of even and odd , respectively). We obtaindirect product decompositions analogous to (1.2), with another countable index set e Σ, P E = Y σ ∈ e Σ P E σ , P O = Y σ ∈ e Σ P O σ , as well as two distinguished cones P E and P O , see Subsection 7.4 (the notation there is abit diﬀerent).Here a new eﬀect arises: in the ﬁrst decomposition, some components are isomorphic to P E and the other are isomorphic to P O , and likewise for the second decomposition. Thusthe two versions, even and odd, are intertwined. But we obtain again a reduction of ourproblem: it suﬃces to study the distinguished cones P E and P O .In Theorem 8.6, we ﬁnd two new branching graphs: Y HL E ( t ) and Y HL O ( t ). They are linkedwith the HL symmetric functions with the negative parameter t ∈ ( − , Y HL E ( t ) are the Young diagrams of even size, and those of Y HL O ( t ) are the Young diagrams ofodd size. The formal edge multiplicities depend on t and are deﬁned from the multiplicationby p in the HL basis. Theorem 8.9 claims that the cones P E and P O are isomorphic to thecones of nonnegative harmonic functions on Y HL E ( − q − ) and Y HL O ( − q − ), respectively.Theorems 8.6 and 8.9 are the main results of the present paper. The construction of Theo-rem 8.6 is unusual in that the formal edge multiplicities are deﬁned through the multiplicationby p instead of p . Another novel phenomenon is the appearance of the Hall–Littlewoodfunctions with negative parameter t (it strongly resembles Ennola’s duality [7], [8], [20]).As a ﬁrst application of the main results, we construct a few examples of invariant measuresincluding an analogue of the Plancherel measure (Section 8.4.1).We can construct a large family of invariant measures by diﬀerent tools (this will be thesubject of a future publication), but at present we do not have a complete classiﬁcation.

In view of the above, Problem 1.1 for the even and odd unitary groups reduces to ﬁndingthe boundaries of the new branching graphs Y HL E ( t ) and Y HL O ( t ) with t = − q − . The latterproblem in turn can be formulated more broadly: Problem 1.3.

Find the boundaries of the even and odd HL-deformed branching graphs Y HL E ( t ) and Y HL O ( t ) with parameter t ∈ ( − , g ∗ to generalized sphericalrepresentations of the semidirect product group G ⋉ ¯ g . (In Lie theory, such semidirectproducts are called Takiﬀ groups .)1.4.

Acknowledgements.

This project started while the ﬁrst author (C. C.) worked atCalifornia Institute of Technology. The research of the second author (G. O.) was supportedby the Russian Science Foundation, project 20-41-09009.

Part GL ( ∞ , q ) -invariant measures The finite general linear group: preliminaries

The group GL ( n, q ) and the Lie algebra gl ( n, q ) . Let q be the power of an oddprime number. Fix the ﬁnite ﬁeld F := F q with q elements. Denote by GL ( n, q ) the groupof invertible n × n matrices with entries in F . Denote by Mat n ( q ) = Mat n ( F ) the associativealgebra of square matrices of order n , over the ﬁnite ﬁeld F ; the corresponding Lie algebra,with commutator [ X, Y ] = XY − Y X , is denoted gl ( n, q ). The group GL ( n, q ) is calledthe general linear group over F , and gl ( n, q ) acts as its Lie algebra. The adjoint action of GL ( n, q ) on gl ( n, q ) is matrix conjugation. The invariant bilinear form ( X, Y ) Tr( XY )on gl ( n, q ) allows one to identify the vector space gl ( n, q ) with its dual, so we may identifythe adjoint and coadjoint actions of GL ( n, q ).Let us agree that GL (0 , q ) := { } is the trivial group and gl (0 , q ) := { } is the zero Liealgebra.2.2. Partitions.

We recall some notions related to integer partitions, which will be usedthroughout the paper.A partition is an inﬁnite sequence λ = ( λ , λ , · · · ) of nonnegative integers such that λ ≥ λ ≥ · · · ≥ λ, µ . The size of the partition λ is | λ | := P i ≥ λ i . Further,we will use the following standard notation (Macdonald [16, Ch. I]): ℓ ( λ ) := max { j | λ j = 0 } , n ( λ ) := X i ≥ ( i − λ i , m i ( λ ) := { j ≥ | λ j = i } , i ≥ . (2.1)Following [16] we identify partitions with their Young diagrams. We denote by Y theset of all partitions (=Young diagrams) and write Y n for the set of partitions of size n . Inparticular, Y only contains the zero partition (= empty Young diagram), to be denoted ∅ .The partition λ ′ = ( λ ′ , λ ′ , · · · ) corresponding to the transposed Young diagram λ ′ is givenby λ ′ k := P j ≥ k m j ( λ ), for all k ≥ CESAR CUENCA AND GRIGORI OLSHANSKI

Nilpotent conjugacy classes in gl ( n, q ) . By a conjugacy class in gl ( n, q ) we meana GL ( n, q )-orbit in this space. A conjugacy class will be called nilpotent if it consists ofnilpotent matrices. The set of nilpotent matrices in gl ( n, q ) will be denoted by Nil( gl ( n, q )).We say that a nilpotent matrix X ∈ Nil( gl ( n, q )) has Jordan type λ ∈ Y n if its Jordanblocks have lengths λ , λ , · · · . This gives a parametrization of nilpotent conjugacy classesin gl ( n, q ) by partitions of size n . The class corresponding to a partition λ ∈ Y n will bedenoted by { λ } . In this notation, the decomposition of Nil( gl ( n, q )) into conjugacy classesis written as Nil( gl ( n, q )) = G λ ∈ Y n { λ } . General conjugacy classes in gl ( n, q ) . Let T n denote the set of conjugacy classes in gl ( n, q ). A matrix X ∈ gl ( n, q )) belonging to a class τ ∈ T n is said to be of type τ .Let NSin( gl ( n, q )) ⊂ gl ( n, q ) be the subset of nonsingular matrices (that is, matrices withnonzero determinant). It is GL ( n, q )-invariant. Let Σ n ⊂ T n be the subset of conjugacyclasses contained in NSin( gl ( n, q )). Elements of Σ n will be called nonsingular classes or nonsingular types . Lemma 2.1.

There is a natural bijection T n ↔ n G s =0 (Σ s × Y n − s ) . (Here we regard Σ as a singleton, so that Σ × Y n is identiﬁed with Y n .) Proof.

It is convenient to regard matrices X ∈ gl ( n, q ) as linear operators on the vector space V := F n . For any X , there is a unique direct sum decomposition V = V ′ ⊕ V such that both V ′ and V are X -invariant, X (cid:12)(cid:12) V ′ is invertible, and X (cid:12)(cid:12) V is nilpotent. Because of uniqueness,this decomposition is GL ( n, q )-invariant. It provides the desired bijection. (cid:3) By virtue of the lemma, each τ ∈ T n is represented by a pair ( σ, λ ), where σ ∈ Σ s and λ ∈ Y n − s for some s , 0 ≤ s ≤ n .Thus, the parametrization of general conjugacy classes in gl ( n, q ) is reduced to the de-scription of the sets Σ s . The latter is given in the remark below, but in fact we do not needit. We will only use the bijection established in the lemma. Remark 2.2 (cf. Macdonald [16], Ch. IV, Sect. 2, or Burgoyne-Cushman [6]) . Let Φ ′ denote the set of irreducible, monic polynomials in F [ x ] with nonzero constant term. Thereis a bijective correspondence between elements of Σ s and maps µ : Φ ′ → Y such that µ ( f ) = ∅ for all but ﬁnitely many polynomials f ∈ Φ ′ and X f ∈ Φ ′ deg( f ) | µ ( f ) | = s, where deg( f ) is the degree of f .3. Invariant measures on L ( q ) : generalities The space L ( q ) . The groups GL ( ∞ , q ) and GLB . Let Mat ∞ ( q ) = Mat ∞ ( F ) be thespace of inﬁnite matrices M = [ m i,j ] i,j ≥ with entries in F . For n ∈ Z ≥ , let L n ( q ) ⊂ Mat ∞ ( q )be the subset of matrices M ∈ Mat ∞ ( q ) such that m i,j = 0, whenever i ≥ j and i > n . Inparticular, L ( q ) is the set of strictly upper triangular matrices. The set L n ( q ) is a vector space over F and, in particular, a commutative group under addition. Note that L n ( q ) iscontained in L n +1 ( q ) as a subgroup of ﬁnite index q n +1 .As an example, a matrix in L ( q ) looks like M = ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ... ∗ ∗ ∗· · · . . . ∈ L ( q ) , where an asterisk stands for an arbitrary element of F . Deﬁnition 3.1.

Let L ( q ) be the inductive limit group lim −→ L n ( q ). As a set, it is the unionof all L n ( q ). In other words, L ( q ) is the set of almost strictly upper triangular matrices.For n ∈ Z ≥ , let L − n ( q ) ⊂ L ( q ) be the subgroup formed by the matrices for which theﬁrst n columns are null. We equip L ( q ) with the topology in which the subgroups L − n ( q )form a fundamental system of neighborhoods of 0. Each L n ( q ) is compact and clopen (bothopen and closed). Therefore the space L ( q ) is locally compact. Moreover, any compactsubset of L ( q ) is contained in some L n ( q ).The natural inclusions GL ( n, q ) ֒ → GL ( n + 1 , q ) give rise to the inductive limit group GL ( ∞ , q ) := lim −→ GL ( n, q ). It is a countable group and it acts on L ( q ) by conjugations.Recall that the group GLB is formed by invertible inﬁnite size matrices over F with ﬁnitelymany entries below the diagonal. Its “Borel subgroup” B ⊂ GLB formed by upper triangularmatrices is a proﬁnite group. As such, it is a compact group in the natural topology. Thistopology is uniquely extended to

GLB from the condition that B is an open subgroup. It iseasily checked that GL ( ∞ , q ) is a dense subgroup of GLB .Like GL ( ∞ , q ), the group GLB acts on L ( q ) by conjugation. We regard this as the coadjoint action (see Section 1.1). Lemma 3.2.

The action

GLB × L ( q ) → L ( q ) is continuous.Proof. It suﬃces to show the similar claim with

GLB replaced by B , and this is a trivialexercise. (cid:3) Radon measures on L ( q ) . For any M ∈ Mat ∞ ( q ), denote M { n } := [ m i,j ] i,j =1 , ··· ,n itsupper-left n × n corner; it is a matrix from gl ( n, q ). An elementary cylinder set of level n ∈ Z ≥ is a subset of the formCyl n ( X ) := { M ∈ L ( q ) | M ∈ L n ( q ) , M { n } = X } ⊂ L ( q ) , (3.1)where n ∈ Z ≥ and X ∈ gl ( n, q ). If we denote by X the unique element from the zero Liealgebra gl (0 , q ), we agree that Cyl ( X ) := L ( q ).More generally, a cylinder set of level n is, by deﬁnition, a (ﬁnite) union of some elementarycylinder sets of the same level. In particular, L n ( q ) is a cylinder set of level n . Note thatany cylinder set of level n is also a cylinder set of level n + 1. In particular, for elementarycylinders we have Cyl n ( X ) = G { Y } Cyl n +1 ( Y ) , where { Y } is a subset of gl ( n + 1 , q ) depending on X ∈ gl ( n, q ). The elementary cylinder sets are clopen compact sets; they form a base of the topology of L ( q ).By a measure we always mean a countably additive nonnegative set function (we will notneed complex or signed measures). Recall that a Radon measure on a locally compact spaceis a Borel measure (possibly of inﬁnite mass) with the property that it takes ﬁnite values oncompact subsets. A Borel measure on L ( q ) is Radon if all sets L n ( q ) have ﬁnite mass. Lemma 3.3.

There is a bijective correspondence between Radon measures on L ( q ) andﬁnitely-additive, nonnegative set functions on the cylinder sets of all levels, with ﬁnite values.Proof. Any Radon measure obviously produces a ﬁnitely-additive function on the cylindersets. Let us show that, conversely, any ﬁnitely-additive function on the ring of cylinder setsadmits a unique extension to a Radon measure. If an extension exists, then it is unique,because the cylinder sets generate the sigma algebra of Borel sets. To show the existence, weare going to apply the Carath´eodory extension theorem from measure theory. Its hypothesisis satisﬁed for the trivial reason that any decomposition of a cylinder set into disjoint opensubsets must be ﬁnite (because all cylinder sets are compact). Thus, the theorem is appli-cable. It gives us the desired countably-additive extension. The latter is a Radon measure,because any compact subset of L ( q ) is contained in some L n ( q ), which is a cylinder set. (cid:3) Invariant measures on L ( q ) .Deﬁnition 3.4. Deﬁne P GL ( ∞ ,q ) as the set of GL ( ∞ , q )-invariant Radon measures on L ( q ). Proposition 3.5.

Any measure P ∈ P GL ( ∞ ,q ) is automatically GLB -invariant.Proof.

Let g ∈ GLB be arbitrary. It suﬃces to prove that for any elementary cylinder set C ⊂ L ( q ), one has P ( gCg − ) = P ( C ). Further, with no loss of generality, it suﬃces toprove this under the additional assumption that n , the level of C , is large enough. We willassume that n is so large that all matrix entries g ij , such that i > j and i > n , are equalto 0. Let h := g { n } be the upper left n × n corner of the matrix g . Our assumption impliesthat h is invertible and hence belongs to GL ( n, q ). Next, the conjugation by g permutes theelementary cylinder sets of level n : if X ∈ gl ( n, q ), then g Cyl n ( X ) g − = Cyl n ( hXh − ) . Because P is GL ( ∞ , q )-invariant, we have P (Cyl n ( hXh − )) = P (Cyl n ( X )). This completesthe proof. (cid:3) Note that P GL ( ∞ ,q ) is a convex cone. Its description will be given in Section 5. In particular,we will describe the extreme measures in the sense of the following deﬁnition. Deﬁnition 3.6.

A measure P ∈ P GL ( ∞ ,q ) is said to be extreme if it is nonzero and lies onsome extreme ray of the cone P GL ( ∞ ,q ) .In other words, a nonzero measure P ∈ P GL ( ∞ ,q ) is extreme if any other measure P ′ ∈P GL ( ∞ ,q ) which is majorated by P is in fact proportional to P . Or else: if P ∈ P GL ( ∞ ,q ) isabsolutely continuous with respect to P , then P ′ is proportional to P .A standard argument shows that P ∈ P GL ( ∞ ,q ) is extreme if and only if it is ergodic in thesense that if A ⊂ L ( q ) is a GL ( ∞ , q )-invariant Borel subset, then is either A is P -null or itscomplement L ( q ) \ A is P -null. See Phelps [19, Proposition 12.4] (although Phelps considersprobability measures, the claim remains valid for inﬁnite measures as well). All nontrivial invariant Radon measures on L ( q ) are inﬁnite. By a trivial mea-sure we mean a multiple of the Dirac measure at the point 0 ∈ L ( q ). Proposition 3.7.

All nontrivial measures P ∈ P GL ( ∞ ,q ) are inﬁnite measures.Proof. Suppose P ∈ P GL ( ∞ ,q ) has ﬁnite total mass; then we show that P is concentrated at { } . Indeed, we may assume that the total mass equals 1. Regard ( L ( q ) , P ) as a probabilityspace. Then the entries m ij of matrices M ∈ L ( q ) turn into F -valued random variables.Consider the collection of random variables { ξ i := m i +1 , : i = 1 , , . . . } . This collectionis exchangeable , because any ﬁnitary permutation of ξ i ’s can be implemented by a suitablepermutation matrix lying in GL ( ∞ , q ). Therefore we may apply de Finetti’s theorem, whichtells us that { ξ i } is a mixture of i.i.d random variables. On the other hand, by the verydeﬁnition of L ( q ), the number of nonzero ξ i ’s is ﬁnite, with P -probability 1. This is onlypossible if all ξ i are in fact equal to 0, with P -probability 1. Using invariance with respectto symmetric group S ( ∞ ) := lim −→ S ( n ), embedded in GL ( ∞ , q ) in a natural way, we seethat m ij = 0 for all i = j , with P -probability 1. Hence P is concentrated on the diagonalmatrices. But such a measure can be GL ( ∞ , q )-invariant only if it is concentrated at 0. (cid:3) Invariant measures on pronilpotent matrices.

Let Nil( L n ( q )) ⊂ L n ( q ) be thesubset of matrices M with nilpotent n × n corners M { n } . Also, letNil( L ( q )) := ∞ [ n =1 Nil( L n ( q ))be the subset of L ( q ) consisting of matrices for which any suﬃciently large corner is nilpotent.Matrices from Nil( L ( q )) will be called pronilpotent matrices . Lemma 3.8.

One has

Nil( L ( q )) = [ g ∈ GL ( ∞ ,q ) g L ( q ) g − . Proof.

Evident, because a similar claim holds in ﬁnite dimension: any nilpotent matrix X ∈ gl ( n, q ) is conjugated to a strictly upper triangular matrix. (cid:3) It follows, in particular, that the set Nil( L ( q )) is open and GL ( ∞ , q )-invariant. (It is GLB -invariant, too.)

Deﬁnition 3.9.

Deﬁne P GL ( ∞ ,q )0 ⊂ P GL ( ∞ ,q ) as the subset of measures which are supportedon Nil( L ( q )). Further, let B GL ( ∞ ,q )0 ⊂ P GL ( ∞ ,q )0 be the subset of measures P that are normal-ized by the condition P ( L ( q )) = 1.Evidently, P GL ( ∞ ,q )0 is a convex cone. Note that its extreme rays are also extreme rays ofthe ambient cone P GL ( ∞ ,q ) . From Lemma 3.8 it follows that if P is a nonzero measure from P GL ( ∞ ,q )0 , then P ( L ( q )) > GL ( ∞ , q ) is countable). This in turnimplies that B GL ( ∞ ,q )0 is a base of the cone P GL ( ∞ ,q )0 .In the next section, we give a description of measures from the cone P GL ( ∞ ,q )0 . Since anynonzero measure from P GL ( ∞ ,q )0 can be uniquely expressed as c · P , for some P ∈ B GL ( ∞ ,q )0 and c >

0, our results also represent a description of measures from B GL ( ∞ ,q )0 . Later, in Section5, we deduce from this a description of the cone P GL ( ∞ ,q ) of general invariant measures on L ( q ). Remark 3.10.

Following [11], denote by U ⊂ B the subgroup formed by the upper-unitriangular matrices. There is a natural bijection L ( q ) → U that assigns to each M ∈ L ( q ) the invertible matrix 1 + M . Given a measure P ∈ B GL ( ∞ ,q )0 , the pushforward of P (cid:12)(cid:12) L ( q ) with respect to the bijection L ( q ) → U is a central probability measure on U in the sense of[11, Deﬁnition 4.3]. In this way we obtain a bijective correspondence between B GL ( ∞ ,q )0 andthe set of central probability measures on U . However, central measures on U catch onlymeasures on pronilpotent matrices and leave aside the more general invariant measures on L ( q ) considered in Section 5. 4. Description of P GL ( ∞ ,q )0 Describing P GL ( ∞ ,q )0 is equivalent to describing the set of nonnegative harmonic functionson certain branching graph. After making this connection, we invoke the recently provedKerov’s conjecture (Matveev [17]) to obtain the desired description (Theorem 4.18).Our construction of ergodic normalized measures supported on pronilpotent matrices es-sentially coincides with the construction of ergodic central measures on U , as given in [11,Theorem 4.6].4.1. Generalities on branching graphs.

We shall consider several branching graphs andthe space of harmonic functions on them. The general notions, at the level of generality thatwe will need, are encapsulated in the following deﬁnition.

Deﬁnition 4.1.

By a branching graph

Γ we mean a graph with graded vertex set F ∞ n =0 Γ n and formal edge multiplicities (or weights ), subject to the following conditions:— all levels Γ n are ﬁnite nonempty sets and Γ is a singleton;— only vertices of adjacent levels can be joined by an edge;— each vertex of level n ∈ Z ≥ is joined with at least one vertex of level n + 1;— each vertex of level n ∈ Z ≥ is joined with at least one vertex of level n − Example 4.2.

Recall the deﬁnition of the

Young graph Y : its vertices are arbitrary parti-tions (=Young diagrams) and the edges are formed by pairs µ ր λ of diagrams, where µ ր λ ⇔ | λ | = | µ | + 1 and µ ⊂ λ. By deﬁnition, all edge weights of Y equal 1. In this section, we will deal with certainbranching graphs which diﬀer from Y by a diﬀerent system of edge weights. In Section 8some new graphs will appear. Deﬁnition 4.3.

Let Γ be a branching graph. Given two vertices, v n ∈ Γ n and v n +1 ∈ Γ n +1 ,forming an edge, let W n +1 n ( v n +1 , v n ) stand for the corresponding weight.(1) A harmonic function on Γ is a real-valued function ϕ on Γ such that ϕ ( v n ) = X v n +1 ∈ Γ n +1 W n +1 n ( v n +1 , v n ) ϕ ( v n +1 ) , v n ∈ Γ n , n ∈ Z ≥ . (2) The set of nonnegative harmonic functions on Γ will be denoted by H + (Γ). Note thatit is a convex cone. (3) Let v denote the root of Γ — the only vertex of level 0. The subset of functions ϕ ∈ H + (Γ) normalized by the condition ϕ ( v ) = 1 will be denoted by H (Γ). Note that it isa convex set, which serves as a base of the cone H + (Γ).(4) The set ex( H (Γ)) of extreme points of the convex set H (Γ) is called the boundary ofΓ. Let R Γ be the space of arbitrary real-valued functions on Γ, equipped with the topologyof pointwise convergence. Proposition 4.4.

The followings claims hold true. (1) H (Γ) is a nonempty compact subset of R Γ . (2) The boundary ex( H (Γ)) ⊂ H (Γ) is a nonempty subset of type G δ , hence a Borel set. (3) There exists a bijective correspondence ϕ ↔ m between functions ϕ ∈ H (Γ) andprobability Borel measures m on the boundary ex( H (Γ)) , given by ϕ ( v ) = Z ψ ∈ ex( H (Γ)) ψ ( v ) m ( dψ ) , v ∈ Γ . This formula also establishes an isomorphism between the cone H + (Γ) and the cone ofﬁnite Borel measures on the boundary. Proof.

Claim (1) is evident. Claim (2) is a well-known fact, see Phelps [19, Proposition 1.3](the fact that the boundary is nonempty follows from the Krein–Milman theorem). Claim(3) can be deduced from Choquet’s theorem [19, Sect. 10]: see, e.g. [18, Theorem 9.2]. (cid:3)

Deﬁnition 4.5.

Let Γ and Γ ′ be two branching graphs with common sets of vertices andedges, but diﬀerent systems of edge weights, { W n +1 n } and { ( W ′ ) n +1 n } . Following Kerov [13,Ch. 1, Sect. 2.2], we say that Γ and Γ ′ are similar if there exists a positive real-valuedfunction f on the vertex set, such that( W ′ ) n +1 n ( v n +1 , v n ) = W n +1 n ( v n +1 , v n ) · f ( v n ) f ( v n +1 ) . (4.1)Then f is called the gauge function from Γ to Γ ′ . With no lost of generality we may (andwill) assume that f ( v ) = 1.The following lemma, although evident, is stated because it will be used several times. Lemma 4.6.

The convex cones of nonnegative harmonic functions on two similar branchinggraphs are aﬃne-isomorphic. More explicitly, if Γ is similar to Γ ′ and f is the gauge functionfrom Γ to Γ ′ , then ϕ ϕ ′ := ϕ · f deﬁnes an aﬃne-isomorphism H + (Γ) ∼ = → H + (Γ ′ ) as wellas an isomorphism of convex sets H (Γ) ∼ = → H (Γ ′ ) . HL-deformed Young graph Y HL ( t ) and its boundary. Recall some terminologyand results from Macdonald [16, Ch. III]. Let t be any real number in the interval (0 , R -algebra of symmetric functions. It is known that Sym is freelygenerated by 1 and the power-sums p , p , · · · . Moreover, Sym has a basis with elementsparametrized by partitions λ ∈ Y , consisting of the Hall-Littlewood functions P λ (; t ). Forconciseness, we shall abbreviate Hall-Littlewood by HL. The HL function P λ (; t ) is homoge-neous of degree | λ | . We also need the Q -version of the HL functions, denoted Q λ (; t ): Q λ (; t ) := b λ ( t ) P λ (; t ) , b λ ( t ) := Y i ≥ Y ≤ j ≤ m i ( λ ) (1 − t j ) . (4.2) The following is the simplest

Pieri rule for HL functions (see [16, Ch. III, (5.7 ′ ), (5.8 ′ )]):(1 − t ) p · Q µ (; t ) = X λ : µ ր λ ψ λ/µ ( t ) Q λ (; t ) , (4.3)where the coeﬃcients ψ λ/µ ( t ) are deﬁned as follows: let k = k ( λ/µ ) be the column numberof the single box that diﬀers λ from µ ; then ψ λ/µ ( t ) := ( − t m k − ( µ ) if k > , k = 1 . (4.4)Observe that all coeﬃcients ψ λ/µ ( t ) are strictly positive.It is convenient for us to rewrite (4.3) in a slightly modiﬁed form: p · Q µ (; t )(1 − t ) | µ | = X λ : µ ր λ ψ λ/µ ( t ) Q λ (; t )(1 − t ) | λ | . (4.5) Deﬁnition 4.7.

The

HL-deformed Young graph Y HL ( t ) with parameter t ∈ (0 ,

1) is thebranching graph with the same vertex and edge sets as in the Young graph Y and with theweights ψ λ/µ ( t ) assigned to the edges µ ր λ .Thus, harmonic functions ϕ on Y HL are deﬁned by the relations ϕ ( µ ) = X λ : µ ր λ ψ λ/µ ( t ) ϕ ( λ ) , µ ∈ Y . (4.6) Remark 4.8.

There is a one-to-one correspondence ϕ ↔ Φ between harmonic functions ϕ ∈ H ( Y HL ( t )) and linear functionals Φ : Sym → R satisfying the conditions • Φ( p F ) = Φ( F ) for any F ∈ Sym (harmonicity); • Φ is nonnegative on the convex cone C HL ( t ) ⊂ Sym spanned by the functions Q λ (; t ), λ ∈ Y (positivity); • Φ(1) = 1 (normalization).This correspondence is given by (compare (4.5) with (4.6)): ϕ ( λ ) = Φ (cid:18) Q λ (; t )(1 − t ) | λ | (cid:19) , λ ∈ Y . We describe the boundary of Y HL ( t ) in the next proposition. To state it, we need a littlepreparation. Let R ≥ be the set of nonnegative real numbers and R ∞≥ be the direct productof countably many copies of R ≥ , equipped with the product topology. Deﬁnition 4.9.

Let Ω( t ) be the set of pairs ω = ( α, β ) ∈ R ∞≥ × R ∞≥ such that α ≥ α ≥ · · · ≥ , β ≥ β ≥ · · · ≥ , ∞ X i =1 α i + (1 − t ) − ∞ X i =1 β i ≤ . Note that Ω( t ) is a compact set.Let C (Ω( t )) be the algebra of continuous functions on Ω( t ) with pointwise operations. Weare going to deﬁne an algebra morphism Sym → C (Ω( t )). Since Sym is freely generated bythe power-sums p , p , · · · , it suﬃces to specify their images — the functions p k ( ω ). We set p ( ω ) ≡ , p k ( ω ) = ∞ X i =1 α ki + ( − k − (1 − t k ) − ∞ X i =1 β ki , k ≥ . (4.7) It is readily checked that the functions p k ( ω ) are continuous. In this way we turn anyelement F ∈ Sym into a continuous function F ( ω ) on Ω( t ). In particular, the HL functions Q λ (; t ) ∈ Sym are turned into continuous functions on Ω, which will be denoted by Q λ ( ω ; t ).Let us call (4.7) the ω -specialization of the algebra Sym.

Remark 4.10 (cf. Bufetov–Petrov [5], Remark 2.6) . Let us associate with each ω = ( α, β ) ∈ Ω( t ) the triple ( α, e β, e γ ), where e γ := 1 − X i α i − (1 − t ) − X i β i ≥ e β is the following double inﬁnite collection of parameters: e β := { β ij : i, j = 1 , , . . . } , β ij := β i t j − (cf. Macdonald [16, Ch. III, §

2, Example 7]). In this notation, (4.7) can be rewritten as p ( ω ) = ∞ X i =1 α i + ∞ X i,j =1 β ij + e γ ; p k ( ω ) = ∞ X i =1 α ki + ( − k − ∞ X i,j =1 β kij , k ≥ . (4.8)The formal specialization (4.8) turns any symmetric function F ∈ Sym into an extendedsymmetric function in the variables ( α, e β, e γ ), in the terminology of Vershik–Kerov [21, Sect.6]. Proposition 4.11 (Matveev) . Recall that t is a ﬁxed number in (0 , and Y HL ( t ) is theHL-deformed Young graph with parameter t . The points in the boundary of Y HL ( t ) areparametrized by the elements ω ∈ Ω( t ) . Speciﬁcally, given ω ∈ Ω( t ) , the correspondingextreme harmonic function ϕ ω ∈ ex( H ( Y HL ( t ))) is given by ϕ ω ( λ ) := Q λ ( ω ; t )(1 − t ) | λ | , λ ∈ Y . This is a special case of Proposition 1.6 in Matveev [17].

Corollary 4.12.

The cone H + ( Y HL ( t )) is isomorphic to the cone of ﬁnite Borel measureson the compact space Ω( t ) . Speciﬁcally, the nonnegative harmonic functions on Y HL ( t ) areprecisely the functions of the form ϕ ( λ ) = 1(1 − t ) | λ | Z ω ∈ Ω( t ) Q λ ( ω ; t ) m ( dω ) , λ ∈ Y , where m is a ﬁnite Borel measure on Ω( t ) .Proof. This follows from Proposition 4.11 combined with Proposition 4.4. A subtle point:we must also make sure that the bijection Ω( t ) → ex( H ( Y HL ( t ))) established in Proposition4.11 is a Borel isomorphism. But this follows from the fact that this map is continuous andhence a homeomorphism, because Ω( t ) is compact. (cid:3) The branching graph Γ GL ( ∞ ,q ) . To each pair of diagrams µ ∈ Y n and λ ∈ Y n +1 , weassign a number L n +1 n ( λ, µ ) as follows. Pick a nilpotent matrix X ∈ Nil( gl ( n, q )) of Jordantype µ and consider the augmented matrices Y ∈ gl ( n + 1 , q ) of the form Y := (cid:20) X x (cid:21) , x ∈ F n . (4.9)6 CESAR CUENCA AND GRIGORI OLSHANSKI

The cone H + ( Y HL ( t )) is isomorphic to the cone of ﬁnite Borel measureson the compact space Ω( t ) . Speciﬁcally, the nonnegative harmonic functions on Y HL ( t ) areprecisely the functions of the form ϕ ( λ ) = 1(1 − t ) | λ | Z ω ∈ Ω( t ) Q λ ( ω ; t ) m ( dω ) , λ ∈ Y , where m is a ﬁnite Borel measure on Ω( t ) .Proof. This follows from Proposition 4.11 combined with Proposition 4.4. A subtle point:we must also make sure that the bijection Ω( t ) → ex( H ( Y HL ( t ))) established in Proposition4.11 is a Borel isomorphism. But this follows from the fact that this map is continuous andhence a homeomorphism, because Ω( t ) is compact. (cid:3) The branching graph Γ GL ( ∞ ,q ) . To each pair of diagrams µ ∈ Y n and λ ∈ Y n +1 , weassign a number L n +1 n ( λ, µ ) as follows. Pick a nilpotent matrix X ∈ Nil( gl ( n, q )) of Jordantype µ and consider the augmented matrices Y ∈ gl ( n + 1 , q ) of the form Y := (cid:20) X x (cid:21) , x ∈ F n . (4.9)6 CESAR CUENCA AND GRIGORI OLSHANSKI Any such matrix Y is nilpotent. By deﬁnition, L n +1 n ( λ, µ ) is the number of those Y ’s thathave Jordan type λ . The deﬁnition is correct because this number obviously does not dependon the choice of X ∈ { µ } .The next proposition provides an explicit formula for L n +1 n ( λ, µ ). Its proof was given inKirillov [14, Sect. 2.3] and Borodin [2, Theorem 2.3]; we present it in Appendix A. Proposition 4.13.

Let n ∈ Z ≥ , µ ∈ Y n , λ ∈ Y n +1 . (i) L n +1 n ( λ, µ ) = 0 unless µ ր λ . (ii) Suppose µ ր λ and denote by k the column number of the single box in λ \ µ . Then L n +1 n ( λ, µ ) = ( q n − P j ≥ k m j ( µ ) (1 − q − m k − ( µ ) ) , if k > ,q n − P j ≥ m j ( µ ) , if k = 1 . (4.10)Recall that the notation m i ( µ ) is explained in (2.1). Note that the ﬁrst expression is alsoapplicable in the case k = 1 if we agree that m ( µ ) = + ∞ and so q − m ( µ ) = 0.An important remark is that L n +1 n ( λ, µ ) > µ ր λ .In the next lemma we connect L n +1 n ( λ, µ ) with the coeﬃcients in the Pieri rule (4.3). Lemma 4.14 (cf. [11], p. 369) . For any Young diagrams µ ր λ we have L n +1 n ( λ, µ ) = ψ λ/µ ( q − ) · q n ( µ ) − n ( n − / q n ( λ ) − ( n +1) n/ , n := | µ | . Recall that the quantity n ( · ) is deﬁned in (2.1). Proof.

Set n := | µ | . From the comparison of (4.10) with (4.4) it follows that the desiredrelation is equivalent to the equality n − X j ≥ k m j ( µ ) ? = (cid:18) n ( µ ) − n ( n − (cid:19) − (cid:18) n ( λ ) − ( n + 1) n (cid:19) , where k is the column number of the unique box in λ \ µ . After simpliﬁcation, the equalityis equivalent to n ( λ ) − n ( µ ) ? = X j ≥ k m j ( µ ) . But this is easily obtained from the deﬁnition of the quantity n ( λ ). (cid:3) Deﬁnition 4.15.

Let Γ GL ( ∞ ,q ) be the branching graph whose vertex and edge sets are thesame as in the Young graph Y , and the weights assigned to the edges µ ր λ are the quantities L n +1 n ( λ, µ ), n = | µ | . Proposition 4.16.

The nonnegative harmonic functions on the graph Γ GL ( ∞ ,q ) are preciselythe functions of the form ϕ GL ( ∞ ,q ) ( λ ) = ϕ ( λ ) · q n ( λ ) − ( | λ | ) , λ ∈ Y , (4.11) where ϕ is an arbitrary nonnegative harmonic function on the graph Y HL ( q − ) .Proof. Lemma 4.14 shows that the graphs Y HL ( q − ) and Γ GL ( ∞ ,q ) are similar in the sense ofDeﬁnition 4.5, with the gauge function from Y HL ( q − ) to Γ GL ( ∞ ,q ) equal to f ( λ ) = q n ( λ ) − ( | λ | ) , λ ∈ Y . (4.12)Then we apply Lemma 4.6. (cid:3) Final result: description of measures.

Any measure P ∈ P GL ( ∞ ,q )0 is uniquelydetermined by its values on the elementary cylinder sets Cyl n ( X ), where X ∈ gl ( n, q ) and n ∈ Z ≥ . Note that P (Cyl n ( X )) = 0 unless X is nilpotent. Lemma 4.17.

There is a bijective correspondence between measures P ∈ P GL ( ∞ ,q )0 and non-negative harmonic functions ϕ GL ( ∞ ,q ) on the branching graph Γ GL ( ∞ ,q ) , uniquely characterizedby the property that for any n ∈ Z ≥ and any nilpotent matrix X ∈ gl ( n, q ) of a given Jordantype µ ∈ Y n , one has P (Cyl n ( X )) = ϕ GL ( ∞ ,q ) ( µ ) . (4.13) Proof.

Let P ∈ P GL ( ∞ ,q )0 . Because P is invariant, the left-hand side of (4.13) only depends onthe Jordan type µ of X . The elementary cylinder Cyl n ( X ) can also be viewed as a cylinderset of level n + 1. As such, it splits into a disjoint union of elementary cylinders Cyl n +1 ( Y ),where Y ranges over the set of matrices of the form (4.9). This entails the equality P (Cyl n ( X )) = X Y P (Cyl n +1 ( Y )) . By the deﬁnition of the quantities L n +1 n ( λ, µ ), among the Y ’s in the sum, exactly L n +1 n ( λ, µ )of them are of Jordan type λ , for each λ ∈ Y n +1 . Therefore, taking (4.13) as the deﬁnitionof a function ϕ GL ( ∞ ,q ) , we see that this function is a nonnegative harmonic function on thegraph Γ GL ( ∞ ,q ) .Conversely, let ϕ GL ( ∞ ,q ) be a nonnegative harmonic function on the graph Γ GL ( ∞ ,q ) . Thenthe above argument shows that formula (4.13) gives rise to a nonnegative, ﬁnitely-additiveset function P on cylinder sets contained in the set of pronilpotent matrices. By virtue ofLemma 3.3, P extends to a true Radon measure on L ( q ), which is supported on the set ofpronilpotent matrices. Its invariance is evident from the very construction.This completes the proof. (cid:3) Combining Lemma 4.17, Proposition 4.16, and Corollary 4.12, we ﬁnally obtain the fol-lowing description of the measures P ∈ P GL ( ∞ ,q )0 . Theorem 4.18. (1)

The measures P ∈ P GL ( ∞ ,q )0 are in one-to-one correspondence with theﬁnite Borel measures m on the compact space Ω( q − ) . (2) Under this correspondence P ↔ m , the mass of an elementary cylinder set Cyl n ( X ) ,where X ∈ Nil( gl ( n, q )) is a nilpotent matrix of Jordan type λ ∈ Y n , is given by P (Cyl n ( X )) = q n ( λ ) − n ( n − / (1 − q − ) n Z ω ∈ Ω( q − ) Q λ ( ω ; q − ) m ( dω ) . (3) The mass P ( L ( q )) is equal to the total mass of m . Thus, the normalized measures P ( i.e. the ones with P ( L ( q )) = 1) correspond to probability measures m . (4) Normalized ergodic measures P correspond to Dirac measures m and hence are parametrizedby points ω ∈ Ω( q − ) . For such measures P = P ω , the above formula reduces to P ω (Cyl n ( X )) = q n ( λ ) − n ( n − / (1 − q − ) n Q λ ( ω ; q − ) , X ∈ { λ } , λ ∈ Y n . (4.14)The last formula coincides with formula (1.3) in Bufetov–Petrov [5], and with (4.3) inGorin–Kerov–Vershik [11]. Examples and remarks.

In this subsection, we use the standard notation( u ; t ) n := n Y j =1 (1 − ut j − ) , ( u ; t ) ∞ := ∞ Y j =1 (1 − ut j − ) . We also employ the notation from Deﬁnition 4.9 and Remark 4.10.We give some examples of normalized ergodic measures P ω , which correspond to certainparticular values of the parameters ω = ( α, β ) ∈ Ω( q − ). We also write down the comple-mentary parameter e γ = 1 − P α i − P (1 − q − ) − β i . To simplify the notation throughoutthis subsection, we set t := q − . Example 4.19 ( α ≡ β = (1 − t, , , . . . ), e γ = 0) . The corresponding normalized ergodicmeasure P ω is the Dirac measure at the point 0 ∈ L ( q ). This is the simplest (and trivial)example. The restriction of P to an elementary cylinder Cyl n ( X ) vanishes unless X = 0 n ,which is the unique matrix of type (1 n ). Example 4.20 ( α = (1 , , , . . . ), β ≡ e γ = 0) . For the corresponding normalized ergodicmeasure P , its restriction to an elementary cylinder Cyl n ( X ) vanishes unless X has type λ = ( n ). The restriction of P ω to L ( q ) admits a nice description: in the matrix coordinates m ij , i < j , this is a product measure, such that each coordinate m i,i +1 does not vanish andis uniformly distributed on F \ { } , while each coordinate m ij with j ≥ i + 2 is uniformlydistributed on F . Example 4.21 ( α = (1 − t, (1 − t ) t, (1 − t ) t , . . . ), β ≡ e γ = 0) . The normalized ergodicmeasure with these parameters will be denoted by P Haar . For this measure, formula (4.14)takes the simple form P Haar (Cyl n ( X )) = t n ( n − / , X ∈ { λ } , λ ∈ Y n , because Q λ (1 , t, t , . . . ; t ) = t n ( λ ) (see [16, Ch. III, §

2, Ex. 1]). From this expression, it isseen that P Haar is the restriction to Nil( L ( q )) of the Haar measure on the additive group L ( q ), with the normalization P Haar ( L ( q )) = 1. The Haar measure on L ( q ) was the subjectof Borodin’s work [2]. Example 4.22 ( α ≡ β ≡ e γ = 1) . The specialization Sym → R with these parametersis the “Plancherel specialization” that sends p to 1 and all other p k ’s to 0. For this reasonwe denote the corresponding normalized ergodic measure by P Planch . Formula (4.14) takesthe form P Planch (Cyl n ( Y )) = t n ( n − − n ( λ ) n ! X λ (1 n ) ( t ) , Y ∈ { λ } , λ ∈ Y n , where X λ (1 n ) ( t ) := h p n , Q λ (; t ) i t and h · , · i t is the HL inner product deﬁned in [16, Ch. III, § § X λ (1 n ) ( t ) are polynomials in t , which can bewritten as certain combinatorial sums ([16, Ch. III, §

7, Ex. 4]).

Remark 4.23 (Sizes of classes { λ } ) . Let λ ∈ Y n . One can show that the number of nilpotentmatrices in the conjugacy class { λ } ⊂ gl ( n, q ) is given by |{ λ }| = ( t ; t ) n t n ( λ ) − n ( n − ( b λ ( t )) − , t = q − . (4.15) Remark 4.24 (The mass of L n ( q )) . Set t = q − , ﬁx ω ∈ Ω( t ), and let P ω be the correspond-ing normalized ergodic measure, as in (4.14). The following formula holds P ω ( L n ( q )) = ( t ; t ) n h n ( ω )(1 − t ) n t n ( n − / . (4.16)Here h n ( ω ), the ω -specialization of h n ∈ Sym, is found from the generating series H ( ω )( z ) := 1 + ∞ X n =1 h n ( ω ) z n = e e γz ∞ Y i =1 ( − β i z ; t ) ∞ − α i z . The proof is based on the fundamental Cauchy identity for the HL functions.

Remark 4.25 (Law of large numbers) . Each normalized ergodic measure P gives rise to asequence { λ ( n ) } of random Young diagrams of growing size n . Bufetov and Petrov [5] proveda law of large numbers for the row and column lengths of λ ( n ). In the case P = P Haar , thiswas done earlier by Borodin [2], together with a central limit theorem.5.

Description of P GL ( ∞ ,q ) In this section we establish an aﬃne isomorphism between the cone P GL ( ∞ ,q ) and a directproduct of countably many copies of the cone P GL ( ∞ ,q )0 (Propositions 5.4 and 5.5). Becausethe structure of P GL ( ∞ ,q )0 was described in Section 4, we obtain in this way a completedescription of invariant Radon measures on L ( q ).5.1. A partition of L ( q ) . As explained in Subsection 2.4, the type of each matrix X ∈ gl ( n, q ) (i.e. the full invariant of its conjugacy class) is represented by a pair τ = ( σ, λ ) ∈ T n ,where σ ∈ Σ m is a nonsingular type and λ ∈ Y n − m is a Young diagram, for some m ≤ n .We are going to partition L ( q ) into countably many invariant subsets L σ ( q ), indexed byelements of the set Σ := ∞ G m =0 Σ m . It is convenient to set | σ | := m if σ ∈ Σ m , and call this number the size of σ .Recall that for an inﬁnite matrix M ∈ L ( q ), we denote its upper-left n × n corner by M { n } . Lemma 5.1.

Fix M ∈ L ( q ) . For each n , let ( σ n , λ ( n ) ) ∈ T n stand for the type of the corner M { n } ∈ gl ( n, q ) . Let n be the minimal number such that M ∈ L n ( q ) . We have σ n = σ n +1 , λ ( n ) ր λ ( n +1) , n ≥ n . Proof.

Suppose n ≥ n and let X := M { n } ∈ gl ( n, q ). Then the corner M { n +1 } is of the form Y = (cid:20) X x (cid:21) , x ∈ F n . Conjugating X by an appropriate matrix from the subgroup GL ( n, q ) ⊂ GL ( n + 1 , q ), wemay assume that X has the block form (cid:20) S N (cid:21) , where S is nonsingular (of type σ n ) and N is nilpotent (of Jordan type λ ( n ) ). Thus, Y can be represented by the 3 × Y = S x N x Next, we can kill x by conjugating Y with an appropriate unitriangular matrix. Indeed, z S x N x − z = S x − Sz N x , and since S is nonsingular, there exists z such that Sz = x .Now the desired result follows from the ﬁrst claim in Proposition 4.13. (cid:3) The lemma shows that for any M ∈ L ( q ), the nonsingular type of the corner M { n } stabi-lizes; let us call it the stable nonsingular type of M and denote it by σ ( M ). The correspon-dence M σ ( M ) gives rise to a map L ( q ) → Σ. We denote its ﬁbres by L σ ( q ): L σ ( q ) := { M ∈ L ( q ) : σ ( M ) = σ } , σ ∈ Σ . Thus, we obtain a partition L ( q ) = G σ ∈ Σ L σ ( q ) . Note that the set Nil( L ( q )) of pronilpotent matrices is one of the parts — it correspondsto the only element of Σ , the empty nonsingular type. Lemma 5.2.

Each L σ ( q ) , σ ∈ Σ , is a nonempty, GL ( ∞ , q ) -invariant, clopen subset.Proof. Fix an arbitrary σ ∈ Σ. From Lemma 5.1 it is seen that for any n ≥ | σ | , theintersection L σ ( q ) with L n ( q ) is a nonempty cylinder set. It follows that L σ ( q ), σ ∈ Σ, isnonempty, GL ( ∞ , q )-invariant, and open. Finally, it is also closed, because its complementis open. (cid:3) The structure of P GL ( ∞ ,q ) .Deﬁnition 5.3. For σ ∈ Σ, let P GL ( ∞ ,q ) σ ⊂ P GL ( ∞ ,q ) be the subset of measures which aresupported on the clopen subset L σ ( q ).Given P ∈ P GL ( ∞ ,q ) and σ ∈ Σ, we denote by P σ the restriction of P to L σ ( q ). It is anelement of P GL ( ∞ ,q ) σ and we can write P = X σ ∈ Σ P σ , P σ ∈ P GL ( ∞ ,q ) σ . Observe that P GL ( ∞ ,q ) σ is a convex cone, for any σ ∈ Σ.Finally, observe that if σ is the only element of Σ , then P GL ( ∞ ,q ) σ coincides with theconvex cone P GL ( ∞ ,q )0 of measures supported on the set of pronilpotent matrices, that wasexamined in Section 4. Proposition 5.4.

The above decomposition determines an aﬃne-isomorphism P GL ( ∞ ,q ) ∼ = Q σ ∈ Σ P GL ( ∞ ,q ) σ .Proof. We only need to check that for any choice of measures P σ ∈ P GL ( ∞ ,q ) σ , σ ∈ Σ, theirsum is a Radon measure. To see this recall that any compact subset of L ( q ) is containedin L n ( q ) for n large enough. On the other hand, L n ( q ) intersects only those sets L σ ( q ) forwhich | σ | ≤ n , and there are only ﬁnitely many such σ ’s. This completes the proof. (cid:3) Proposition 5.5.

For any σ ∈ Σ , there is an aﬃne-isomorphism P GL ( ∞ ,q )0 ∼ = → P GL ( ∞ ,q ) σ ofconvex cones. Proof.

Fix σ ∈ Σ and let s := | σ | . We may assume that s >

0, because if s = 0, then σ is theunique element of Σ and P GL ( ∞ ,q ) σ = P GL ( ∞ ,q )0 . Let µ ∈ Y n and λ ∈ Y n +1 . Pick a matrix X = (cid:20) S N (cid:21) ∈ gl ( s + n, q ) , where S ∈ gl ( s, q ) is a nonsingular matrix of type σ and N ∈ gl ( n, q ) is a nilpotent matrixof Jordan type µ ∈ Y n .Denote by L n +1 n ( λ, µ | σ ) the number of column vectors x ∈ F s + n such that the matrix Y := (cid:20) X x (cid:21) = S x N x ∈ gl ( s + n + 1 , q )has type ( σ, λ ). This number does not depend on the choice of X .From the proof of Lemma 5.1 we see that L n +1 n ( λ, µ | σ ) = q s L n +1 n ( λ, µ ) , (5.1)where the quantity L n +1 n ( λ, µ ) is deﬁned in Subsection 4.3. Recall that L n +1 n ( λ, µ ) is nonzeroprecisely when µ ր λ .Let Γ GL ( ∞ ,q ) σ denote the branching graph with the same vertices and edges as in the Younggraph, and with the edge weights L n +1 n ( λ, µ | σ ). The same argument as in Lemma 4.17yields an isomorphism between the cone P GL ( ∞ ,q ) σ and the cone of nonnegative harmonicfunctions on the graph Γ GL ( ∞ ,q ) σ .On the other hand, from the relation (5.1), it is seen that the graph Γ GL ( ∞ ,q ) σ is similar(in the sense of Deﬁnition 4.5) to the graph Γ GL ( ∞ ,q ) . We know that nonnegative harmonicfunctions on Γ GL ( ∞ ,q ) correspond to measures P ∈ P GL ( ∞ ,q )0 (Lemma 4.17). This yields thedesired isomorphism. (cid:3) Part U (2 ∞ , q ) - and U (2 ∞ + 1 , q ) -invariant measures Conjugacy classes in skew-Hermitian matrices

The unitary group over a ﬁnite ﬁeld.

Denote e F := F q ⊃ F , and let F : e F → e F bethe Frobenius map deﬁned by F ( x ) := x q . It is the non-trivial involutive automorphism of e F that ﬁxes F . We denote x := F ( x ). Fix ǫ ∈ e F \ F such that ǫ = − ǫ ; then a + bǫ = a − bǫ ,for any a, b ∈ F . For ﬁnite ﬁelds, the Frobenius map x x plays the role that conjugationplays for C , and ǫ plays the role of the imaginary unit i . In the same spirit, the conjugatetranspose A ∗ of a rectangular matrix A with entries in e F is obtained by transposing A andapplying the Frobenius map to each entry.Let E be a ﬁnite-dimensional space over e F ; by a sesquilinear form on E we mean a map τ : E × E → e F which is additive in each variable and satisﬁes τ ( ax, y ) = aτ ( x, y ) , τ ( x, ay ) = aτ ( x, y ) , x, y ∈ E, a ∈ e F . The sesquilinear form τ is Hermitian if τ ( x, y ) = τ ( y, x ) , x, y ∈ E. All nondegenerate sesquilinear Hermitian forms on a ﬁxed vector space are equivalent, seee.g. [4, Theorem 4.1] or [15, Sect. 4]. A linear operator A : E → E is said to be τ -skew-Hermitian if τ ( Au, v ) + τ ( u, Av ) = 0 , u, v ∈ E, (6.1)whereas A is said to be τ -Hermitian if τ ( Au, v ) = τ ( u, Av ) , u, v ∈ E. (6.2)To denote the dimension of vector spaces and the size of matrices, we will use the symbol N , because later we will need to distinguish between the cases of even N = 2 n and odd N = 2 n + 1 dimensions.When E = e F N is the vector space of N -column vectors, then a nondegenerate sesquilinearform τ on E corresponds to a nonsingular N × N matrix T via the relation: τ ( x, y ) = y ∗ T x.

Evidently, τ is Hermitian iﬀ T ∗ = T .On the coordinate space e F N , let us ﬁx the nondegenerate sesquilinear Hermitian form τ : e F N × e F N → e F corresponding to the matrix W N with 1’s in the secondary diagonal and0’s elsewhere: W N := . . . . (6.3)In other words, if e , . . . , e N are the vectors from the canonical basis of e F N , then the form τ is given by τ ( e r , e s ) = δ r,N +1 − s , r, s = 1 , . . . , N. (6.4)Given a matrix A ∈ Mat N ( e F ), we say that A is τ -skew-Hermitian , or τ -Hermitian , if thesame is true of the corresponding linear operator A : e F N → e F N (see (6.1) and (6.2)). Moredirectly, if we deﬁne the τ -conjugate of A by A ♯ := W N A ∗ W N , then A is τ -skew-Hermitianif A ♯ = − A , and A is τ -Hermitian if A ♯ = A . For simplicity, we call a matrix A ∈ Mat N ( e F )simply skew-Hermitian if A ♯ = − A , or Hermitian if A ♯ = A .The unitary group U ( N, q ) is the subgroup of matrices in GL ( N, q ) = GL ( N, e F ) thatpreserve τ : U ( N, q ) := { g ∈ GL ( N, q ) | τ ( gu, gv ) = τ ( u, v ) , for all u, v ∈ e F N } . The Lie algebra of U ( N, q ), to be denoted u ( N, q ), is the Lie subalgebra of gl ( N, q ) = gl ( N, e F ) consisting of the skew-Hermitian matrices: u ( N, q ) := { A ∈ gl ( N, q ) | τ ( Au, v ) + τ ( u, Av ) = 0 , for all u, v ∈ e F N } . The group U ( N, q ) acts on u ( N, q ) by matrix conjugation. As in the case of GL ( N, q ), thisadjoint action may be identiﬁed with the coadjoint action.Denote the set of Hermitian matrices byHerm(

N, q ) := { A ∈ gl ( N, q ) | τ ( Au, v ) = τ ( u, Av ) , for all u, v ∈ e F N } . It is also a U ( N, q )-module, with action given by conjugation. Since Herm( N, q ) = ǫ · u ( N, q ), the U ( N, q )-modules Herm( N, q ) and u ( N, q ) are equivalent. We can switchfrom one space to the other if necessary.6.2. Nilpotent conjugacy classes in u ( N, q ) . By a conjugacy class in u ( n, q ) we meana U ( N, q )-orbit. Let Nil( u ( N, q )) ⊆ u ( N, q ) denote the subset of nilpotent matrices. It is U ( N, q )-invariant, so we may speak of nilpotent conjugacy classes in u ( N, q ) just as we didfor gl ( n, q ). We need the following well-known result. Proposition 6.1.

Nilpotent conjugacy classes in u ( N, q ) are parametrized by partitions λ ∈ Y N — exactly as in the case of gl ( N, q ) .Proof. The Jordan type of a nilpotent matrix is obviously an invariant of its conjugacy class.Switching to τ -Hermitian matrices, we have to check two claims:(1) for each λ ∈ Y N , there exists a matrix X ∈ Herm(

N, q ) with this Jordan type;(2) if two matrices X, Y ∈ Herm(

N, q )) are of the same Jordan type, then they areconjugated by an element of the subgroup U ( N, q ) ⊆ GL ( N, q ).To show (1) we exhibit a concrete model for X .Consider the N -dimensional vector space V over e F with the distinguished basis { v ij } in-dexed by the boxes ( i, j ) of the Young diagram λ . Let X : V → V be the operator deﬁned by Xv ij = v i,j − , with the understanding that v i, := 0. This operator is nilpotent and of Jordantype λ . Next, let τ be the sesquilinear form on V with the property that τ ( v ij , v i,λ i − j +1 ) = 1and all other scalar products are 0. Clearly, τ is nondegenerate and Hermitian, and the oper-ator X is τ -Hermitian. Because all nondegenerate sesquilinear Hermitian forms of dimension N are equivalent, this gives the desired result.To show (2), we use the fact that if two matrices X, Y ∈ Herm(

N, q ) are conjugated byan element of GL ( N, q ), then they are also conjugated by an element of U ( N, q ). This iseasily proved by using the Lang–Steinberg theorem, see the expository paper by Springerand Steinberg in [1], Section E, Example 3.5 (a) (this example concerns conjugacy classes inthe group U ( N, q ), but the same argument works for its Lie algebra u ( N, q ) or the spaceHerm( N, q )). (cid:3) Proposition 6.2.

Each nilpotent conjugacy class in u ( N, q ) contains a strictly upper tri-angular matrix. Note that this is not evident, in contrast to the case of gl ( n, q ). Proof.

In invariant terms, the claim is equivalent to the following. Let V be a vector spaceover e F , of dimension 2 m or 2 m + 1, τ be a nondegenerate sesquilinear form on V , and X : V → V be a nilpotent, τ -skew-Hermitian operator. Then there exists a complete ﬂag { V i } , which is preserved by X and has the form { } ⊂ V ⊂ · · · ⊂ V m ⊂ V ⊥ m − ⊂ · · · ⊂ V ⊥ ⊂ V, for dim V = 2 m , { } ⊂ V ⊂ · · · ⊂ V m ⊂ V ⊥ m ⊂ V ⊥ m − ⊂ · · · ⊂ V ⊥ ⊂ V, for dim V = 2 m + 1 , where the ﬁrst m subspaces are τ -isotropic and the symbol ( · · · ) ⊥ means orthogonal com-plement.We prove this by induction on dim V . If X = 0, there is nothing to prove, so we assume X = 0. Since X is nilpotent, its kernel ker X is nonzero. We claim that ker X containsa nonzero isotropic vector v unless dim V = 1. Indeed, if dim(ker X ) ≥

2, then this holds true because any subspace of dimension ≥ X ) = 1 and ker X is non-isotropic, then we have the orthogonal decomposition V = ker X ⊕ (ker X ) ⊥ , which is also X -invariant, but then (ker X ) ⊥ must be null andhence dim V = 1, because otherwise X would have a nontrivial kernel in (ker X ) ⊥ , which isimpossible.The case dim V = 1 being trivial, we assume dim V ≥

2. Then we take as V the one-dimensional isotropic subspace spanned by v . If dim V = 2, we are done. If dim V >

2, thenthe subspace V ⊥ is strictly larger than V . Since it is X -invariant (here we use the fact that X is τ -skew-Hermitian), our task is reduced to the quotient space V ⊥ /V . This argumentyields the desired induction step. (cid:3) General conjugacy classes in u ( N, q ) . The arguments of Subsection 2.4 are ex-tended to the case of u ( N, q ) with evident minor modiﬁcations.We denote by e T N the set of conjugacy classes in u ( N, q ). A matrix X ∈ u ( N, q ) belongingto a class τ ∈ e T N is said to be of type τ .Let NSin( u ( N, q )) ⊂ u ( N, q ) be the subset of nonsingular matrices. It is U ( N, q )-invariant. Let e Σ N ⊂ e T N be the subset of conjugacy classes contained in NSin( u ( N, q )).Elements of e Σ N will be called nonsingular classes or nonsingular types . Lemma 6.3.

There is a natural bijection e T N ↔ N G s =0 ( e Σ s × Y N − s ) . (Here we regard e Σ as a singleton, so that e Σ × Y N is identiﬁed with Y N .) Proof.

We argue as in the proof of Lemma 2.1. In invariant terms, the set e T N can be identiﬁedwith the set of equivalence classes of triples ( V, τ, X ), where V is an N -dimensional vectorspace over e F , τ is a nondegenerate sesquilinear Hermitian form on V , and X : V → V isa τ -skew-Hermitian operator. We consider again the canonical X -invariant decomposition V = V ′ ⊕ V with the property that X (cid:12)(cid:12) V ′ is nonsingular and X (cid:12)(cid:12) V is nilpotent, and observethat it is orthogonal with respect to τ . This gives us two invariants: the type of X (cid:12)(cid:12) V ′ andthe Jordan type of X (cid:12)(cid:12) V , which determine the equivalence class of ( V, τ, X ) uniquely. In thisway we obtain an embedding e T N ֒ → F Ns =0 ( e Σ s × Y N − s ). Finally, this map is also obviouslysurjective, which leads to the desired bijection. (cid:3) Thus, the parametrization of general conjugacy classes in u ( N, q ) is reduced to the explicitdescription of the nonsingular types. For completeness, we give it in the remark below, butin fact we do not use it in the present paper. Remark 6.4 (cf. Remark 2.2) . Let f ( x ) ∈ e F [ x ] be a nonconstant, monic polynomial withnonzero constant coeﬃcient: f ( x ) = x d + c x d − + · · · + c d − x + c d , where d = deg( f ) ≥ c d = 0 . (6.5)Deﬁne e f ( x ) := ( − deg( f ) · f ( − x ), that is, e f ( x ) = x d + e c x d − + · · · + e c d − x + e c d , e c i := ( − i c i . The roots of the polynomial e f ( x ) (in the algebraic closure of e F ) are the result of applyingthe map x

7→ − F ( x ) = − x = − x q to the roots of f ( x ).We say that f is almost-irreducible if its roots form a single orbit under the map x F ( x ). More explicitly, a polynomial f of the form (6.5) is almost-irreducible if either • f is irreducible in e F [ x ] and f = e f (deg( f ) is odd in this case), or • f = g e g , where g is irreducible in e F [ x ] and g = e g (deg( f ) is even in this case).Denote by e Φ ′ the set of almost-irreducible polynomials in e F [ x ]. There is a bijective cor-respondence between elements of e Σ s and maps e µ : e Φ ′ → Y such that e µ ( f ) = ∅ for all butﬁnitely many polynomials f ∈ e Φ ′ and X f ∈ e Φ ′ deg( f ) | e µ ( f ) | = s, where deg( f ) is the degree of f . This result can be extracted from [24] or [7].7. Invariant measures on skew-Hermitian matrices

We need to diﬀerentiate between unitary groups of even and odd dimension.It is convenient to index rows and columns of matrices in the even unitary group U (2 n, q )by the (2 n )-element set {− n, . . . , − , , . . . , n } . This leads to the inclusions U (2 n, q ) ֒ → U (2 n + 2 , q ), and therefore to the inductive limit U (2 ∞ , q ) := lim −→ U (2 n, q ). It is naturallya group of inﬁnite size matrices of format ( Z \ { } ) × ( Z \ { } ). The group U (2 ∞ , q ) iscountable, in fact, the ( i, j )-entry of a matrix from U (2 ∞ , q ) equals δ i,j , for all but ﬁnitelymany entries.Likewise, index rows and columns of matrices in the odd unitary group U (2 n + 1 , q )by the (2 n + 1)-element set {− n, . . . , , . . . , n } . This leads to the inductive limit group U (2 ∞ + 1 , q ) := lim −→ U (2 n + 1 , q ), which is also countable and whose elements are inﬁnitesize matrices of format Z × Z .The superscripts E and O make reference to even and odd , respectively. Both U (2 ∞ , q )and U (2 ∞ + 1 , q ) are called inﬁnite unitary groups .7.1. The group U (2 ∞ , q ) and the space L E ( q ) . The inﬁnite size matrices in this sub-section are of format ( Z \ { } ) × ( Z \ { } ) — they are now two-sided inﬁnite matrices. LetMat E ∞ ( q ) be the space of matrices M = [ m i,j ] i,j ∈ Z \{ } with entries in e F . For n ∈ Z ≥ , let L E n ( q ) ⊂ Mat E ∞ ( q ) be the subset of matrices M such that: • m i,j = 0 whenever i ≥ j , and max {| i | , | j |} > n ; • M is skew-Hermitian, i.e. m − b, − a = − m a,b (= − m qa,b ) for all a, b ∈ Z \ { } . In particular, m − k,k ∈ ǫ · F for all k ∈ Z \ { } .For each n ∈ Z ≥ , the set L E n ( q ) is a vector space over F , therefore a commutative additivegroup. The set L E ( q ) consists of strictly upper triangular skew-Hermitian matrices of format( Z \ { } ) × ( Z \ { } ).6 CESAR CUENCA AND GRIGORI OLSHANSKI

We need to diﬀerentiate between unitary groups of even and odd dimension.It is convenient to index rows and columns of matrices in the even unitary group U (2 n, q )by the (2 n )-element set {− n, . . . , − , , . . . , n } . This leads to the inclusions U (2 n, q ) ֒ → U (2 n + 2 , q ), and therefore to the inductive limit U (2 ∞ , q ) := lim −→ U (2 n, q ). It is naturallya group of inﬁnite size matrices of format ( Z \ { } ) × ( Z \ { } ). The group U (2 ∞ , q ) iscountable, in fact, the ( i, j )-entry of a matrix from U (2 ∞ , q ) equals δ i,j , for all but ﬁnitelymany entries.Likewise, index rows and columns of matrices in the odd unitary group U (2 n + 1 , q )by the (2 n + 1)-element set {− n, . . . , , . . . , n } . This leads to the inductive limit group U (2 ∞ + 1 , q ) := lim −→ U (2 n + 1 , q ), which is also countable and whose elements are inﬁnitesize matrices of format Z × Z .The superscripts E and O make reference to even and odd , respectively. Both U (2 ∞ , q )and U (2 ∞ + 1 , q ) are called inﬁnite unitary groups .7.1. The group U (2 ∞ , q ) and the space L E ( q ) . The inﬁnite size matrices in this sub-section are of format ( Z \ { } ) × ( Z \ { } ) — they are now two-sided inﬁnite matrices. LetMat E ∞ ( q ) be the space of matrices M = [ m i,j ] i,j ∈ Z \{ } with entries in e F . For n ∈ Z ≥ , let L E n ( q ) ⊂ Mat E ∞ ( q ) be the subset of matrices M such that: • m i,j = 0 whenever i ≥ j , and max {| i | , | j |} > n ; • M is skew-Hermitian, i.e. m − b, − a = − m a,b (= − m qa,b ) for all a, b ∈ Z \ { } . In particular, m − k,k ∈ ǫ · F for all k ∈ Z \ { } .For each n ∈ Z ≥ , the set L E n ( q ) is a vector space over F , therefore a commutative additivegroup. The set L E ( q ) consists of strictly upper triangular skew-Hermitian matrices of format( Z \ { } ) × ( Z \ { } ).6 CESAR CUENCA AND GRIGORI OLSHANSKI For instance, a matrix from L E ( q ) looks as follows: M = . . . . . . ∗ ∗ ∗ ∗ ∗ ∗ ⋆ ∗ ∗ ∗ ∗ ⋆ ∗ m − , − m − , − m − , m − , ∗ ∗ m − , − m − , − m − , m − , ∗ ∗ m , − m , − m , m , ∗ ∗ m , − m , − m , m , ∗ ∗ ∗ . . . . . . ∈ L E ( q );the 4 × m i,j ] i,j ∈{− , − , , } belongs to u (4 , q ), an asterisk above the secondarydiagonal stands for an arbitrary element from e F , an asterisk with a bar below the secondarydiagonal means that those elements are determined (conjugate and multiply by −

1) by thoseabove the diagonal, and a star on the secondary diagonal stands for an element from ǫ · F .Let L E ( q ) be the inductive limit group lim −→ L E n ( q ) arising from the natural inclusions L E n ( q ) ֒ → L E n +1 ( q ). As a set, L E ( q ) consists of the almost strictly upper triangular skew-Hermitian matrices of format ( Z \ { } ) × ( Z \ { } ). For n ∈ Z ≥ , let L E − n ( q ) ⊂ L E ( q ) bethe subgroup consisting of those matrices M ∈ L E ( q ) for which m i,j = 0, whenever | i | ≤ n and | j | ≤ n . We equip the group L E ( q ) with the topology in which the subgroups L E − n ( q )form a fundamental system of neighborhoods of 0.Each L E n ( q ) is compact and clopen, so L E ( q ) is locally compact. Any compact subset of L E ( q ) is contained in some L E n ( q ).The group U (2 ∞ , q ) acts on L E ( q ) by conjugation; this action preserves the topology of L E ( q ).For any M ∈ Mat E ∞ ( q ), denote by M [ n ] := [ m i,j ] i,j = − n, ··· , − , , ··· ,n its central (2 n ) × (2 n )submatrix. If M ∈ L E n ( q ), then M [ n ] belongs to u (2 n, q ). A basis for the topology of L E ( q )is given by the elementary cylinder setsCyl E n ( X ) := { M ∈ L E ( q ) | M ∈ L E n ( q ) , M [ n ] = X } , n ∈ Z ≥ , X ∈ u (2 n, q ) . If X is the unique element of the zero Lie algebra u (0 , q ), we agree that Cyl E ( X ) = L E ( q ).Let Nil( L E n ( q )) ⊂ L E n ( q ) be the subset of matrices M = [ m i,j ] i,j ∈ Z \{ } for which itssubmatrix M [ n ] is nilpotent. Equivalently, Nil( L E n ( q )) is the union of cylinder sets Cyl E n ( X ),where X ranges over all nilpotent matrices from u (2 n, q ). Also letNil( L E ( q )) := ∞ [ n =0 Nil( L E n ( q )) ⊂ L E ( q ) . Matrices from Nil( L E ( q )) are called pronilpotent matrices. Proposition 6.2 shows thatNil( L E ( q )) = [ u ∈ U (2 ∞ ,q ) ( u · L E ( q ) · u − ) . As a result, Nil( L E ( q )) is clopen and U (2 ∞ , q )-invariant. The group U (2 ∞ + 1 , q ) and the space L O ( q ) . In this subsection, our inﬁnite sizematrices are of format Z × Z . Deﬁne the spaces Mat O ∞ ( q ), L O n ( q ) ( n ∈ Z ) and L O ( q ) =lim −→ L O n ( q ), in exactly the same way as in the previous subsection, with the only diﬀerencethat we replace E by O everywhere.For instance, a matrix from L O ( q ) looks as follows: M = . . . . . . ∗ ∗ ∗ ∗ ∗ ⋆ ∗ ∗ ∗ ⋆ ∗ m − , − m − , m − , ∗ ∗ m , − m , m , ∗ ∗ m , − m , m , ∗ ∗ ∗ . . . . . . ∈ L O ( q );here, the 3 × m i,j ] i,j ∈{− , , } belongs to u (3 , q ) (the asterisks, stars and asteriskswith a bar have the same meaning as in the example from the previous subsection).As before, deﬁne the topology on L O ( q ) by declaring that the subgroups L E − n ( q ) ( n ∈ Z ≥ ) form a fundamental system of neighborhoods of 0. With respect to this topology, each L O n ( q ) is compact and clopen, so L O ( q ) is locally compact. Moreover, the conjugationaction of U (2 ∞ + 1 , q ) on L O ( q ) preserves the topology.For any M ∈ Mat O ∞ ( q ), denote by M [ n ] := [ m i,j ] i,j = − n,..., ,...,n its central (2 n + 1) × (2 n + 1)submatrix. If M ∈ L O n ( q ), then M [ n ] ∈ u (2 n + 1 , q ). The elementary cylinder sets, whichform a basis for the topology of L O ( q ), areCyl O n ( X ) := { M ∈ L O ( q ) | M ∈ L O n ( q ) , M [ n ] = X } , n ∈ Z ≥ , X ∈ u (2 n + 1 , q ) . Also deﬁne Nil( L O n ( q )) ( n ∈ Z ≥ ) and Nil( L O ( q )) as in the previous subsection, byreplacing E by O everywhere. The set Nil( L O ( q )) is clopen and U (2 ∞ + 1 , q )-invariant.7.3. Invariant Radon measures.Deﬁnition 7.1.

Deﬁne P U (2 ∞ ,q ) as the convex cone of U (2 ∞ , q )-invariant Radon mea-sures on L E ( q ). Likewise, P U (2 ∞ +1 ,q ) is the convex cone of U (2 ∞ + 1 , q )-invariant Radonmeasures on L O ( q ).Let G stand for any of the groups U (2 ∞ , q ), U (2 ∞ + 1 , q ). As pointed out in Section1, one can deﬁne a GLB -type topological completion G ⊃ G . The completion G is deﬁnedas the group of inﬁnite size matrices M with ﬁnitely many entries below the diagonal andsuch that M W ∞ M ∗ = W ∞ , where W ∞ is the inﬁnite size matrix with 1’s in the secondarydiagonal and 0’s elsewhere, and M ∗ denotes the conjugate transpose of M . The topologyon G is uniquely deﬁned by the condition that the subgroup of upper triangular matricescontained in G (which is a proﬁnite group) is an open subgroup.The matrices in the topological completion G ⊃ G are of format ( Z \ { } ) × ( Z \ { } ), if G = U (2 ∞ , q ), and of format Z × Z , if G = U (2 ∞ + 1 , q ). We denote these topologicalcompletions by UB E ⊃ U (2 ∞ , q ) and UB O ⊃ U (2 ∞ + 1 , q ).The groups UB E and UB O act by conjugation on L E ( q ) and L O ( q ), respectively. Theseare inﬁnite-dimensional versions of the coadjoint action, as discussed in Section 1. The action maps UB E × L E ( q ) → L E ( q ) , UB O × L O ( q ) → L O ( q ) , are continuous. Measures from P U (2 ∞ ,q ) and P U (2 ∞ +1 ,q ) are automatically invariant underthe action of the larger groups UB E and UB O , respectively: the proof is the same as inProposition 3.5. Deﬁnition 7.2.

Let P U (2 ∞ ,q )0 ⊂ P U (2 ∞ ,q ) be the subset (which is also a convex cone) ofmeasures which are supported on Nil( L E ( q )). Likewise, P U (2 ∞ +1 ,q )0 ⊂ P U (2 ∞ +1 ,q ) is theconvex cone of U (2 ∞ + 1 , q )-invariant Radon measures supported on Nil( L O ( q )). Remark 7.3 (cf. Remark 3.10) . By virtue of Proposition 6.2, nontrivial measures P in P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 can be normalized so that P ( L E ) = 1 and P ( L O ) = 1, respectively.Their restrictions to L E and L O can be characterized as central probability measures — thedeﬁnition is easily adapted from [11, Deﬁnition 4.3].7.4. Relationship among P U (2 ∞ ,q ) , P U (2 ∞ +1 ,q ) , P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 . This sub-section is an analogue of Subsection 5.2 — we outline how the structures of P U (2 ∞ ,q ) and P U (2 ∞ +1 ,q ) can be understood from P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 . We omit proofs because theycopy the arguments from Subsection 5.2.Recall (see Subsection 6.3) our notation e T N for the set of conjugacy classes in u ( N, q ) andthe decomposition e T N = F Ns =0 ( e Σ s × Y N − s ), where e Σ s denotes the set of nonsingular classesin u ( s, q ). Denote e Σ := ∞ G m =0 e Σ m , and whenever σ ∈ e Σ s , we set | σ | := s .For any matrix M from L E ( q ) or L O ( q ), an analogue of Lemma 5.1 holds: the nonsingularcomponent σ n of the submatrix M [ n ] stabilizes as n gets large; we call it the stable nonsingulartype and denote it by σ ( M ); in this way, we obtain two maps L E ( q ) → e Σ and L O ( q ) → e Σ.Setting L E ,σ ( q ) := { M ∈ L E ( q ) : σ ( M ) = σ } , L O ,σ ( q ) := { M ∈ L O ( q ) : σ ( M ) = σ } , σ ∈ e Σ , we obtain the partitions L E ( q ) = G σ ∈ e Σ L E ,σ ( q ) , L O ( q ) = G σ ∈ e Σ L O ,σ ( q ) , into nonempty invariant clopen subsets.This in turn entails aﬃne-isomorphisms of convex cones P U (2 ∞ ,q ) ∼ = Y σ ∈ e Σ P E σ , P U (2 ∞ +1 ,q ) ∼ = Y σ ∈ e Σ P O σ , where, by deﬁnition, P E σ ⊂ P U (2 ∞ ,q ) is formed by the measures supported on L E ,σ ( q ) ⊂ L E ( q ), and P O σ ⊂ P U (2 ∞ +1 ,q ) is formed by the measures supported on L O ,σ ( q ) ⊂ L O ( q ).Thus, the description of P U (2 ∞ ,q ) and P U (2 ∞ +1 ,q ) is reduced to the description of thecones P E σ and P O σ , where σ ranges over e Σ.Note that if σ is the unique element of e Σ , the corresponding invariant clopen subsets L E ,σ ( q ) and L O ,σ ( q ) are precisely the sets of pronilpotent matrices, so the corresponding cones coincide with the cones P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 , respectively. The ﬁnal result is ananalogue of Proposition 5.5. Proposition 7.4.

For any σ ∈ e Σ , the following aﬃne-isomorphisms of convex cones hold P E σ ∼ = ( P U (2 ∞ ,q )0 , for | σ | even , P U (2 ∞ +1 ,q )0 , for | σ | odd; P O σ ∼ = ( P U (2 ∞ +1 ,q )0 , for | σ | even , P U (2 ∞ ,q )0 , for | σ | odd . The conclusion is that the structures of P U (2 ∞ ,q ) and P U (2 ∞ +1 ,q ) are determined by P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 . Finally, the study of the latter convex cones is undertaken in thenext section.8. The branching graphs of P U (2 ∞ ,q )0 , P U (2 ∞ +1 ,q )0 and Ennola’s duality The main results of this section are two theorems.Theorem 8.6 contains a simple computation with HL functions; its signiﬁcance is thatit implies the existence of new branching graphs, which are based on HL functions with negative parameter t . We call them the even and odd HL-deformed Young graphs.Theorem 8.9 translates the problem of characterizing P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 into theproblem of describing the convex cone of nonnegative harmonic functions on these graphs.As applications, we exhibit examples of measures belonging to P U (2 ∞ ,q )0 and P U (2 ∞ +1 ,q )0 .8.1. The branching graphs Γ U (2 ∞ ,q ) and Γ U (2 ∞ +1 ,q ) . Let n ∈ Z ≥ be arbitrary, and set N = 2 n or N = 2 n + 1. Let µ ∈ Y N and X ∈ Nil( u ( N, q )) be nilpotent of Jordan type µ .Let λ ∈ Y N +2 and denote by e L n +1 n ( λ, µ ) the number of (nilpotent, skew-Hermitian) matricesof the form Y := − x ∗ W N ǫy X x , x ∈ e F N , y ∈ F , (8.1)which are of type λ . Recall that W N ∈ Mat N ( e F ) was deﬁned in (6.3). Evidently, the quantity e L n +1 n ( λ, µ ) does not depend on the speciﬁc choice of matrix X of Jordan type µ .Our notation e L n +1 n ( λ, µ ) is somewhat imprecise because it is unclear whether | µ | = 2 n , | λ | = 2 n + 2, or | µ | = 2 n + 1, | λ | = 2 n + 3. However, this will not cause any issue. Deﬁnition 8.1.

Let µ ∈ Y N and λ ∈ Y N +2 , where N ≥

0. We write µ րր λ if µ ⊂ λ and the two boxes of the skew diagram θ := λ \ µ either lie in a single column (so that θ isa vertical domino) or lie in two consecutive columns (in particular, θ may be a horizontaldomino).The following result gives explicit formulas for e L n +1 n ( λ, µ ). This is an analogue of Propo-sition 4.13, but the proof is more laborious; it is deferred to Appendix A. Proposition 8.2.

Let n ∈ Z ≥ , and set N = 2 n or N = 2 n + 1 . Let µ ∈ Y N , λ ∈ Y N +2 bearbitrary. Recall the notation m i ( µ ) from (2.1) . (i) The quantity e L n +1 n ( λ, µ ) is nonzero if and only if µ րր λ . (ii) Suppose µ րր λ . The value of e L n +1 n ( λ, µ ) depends on the columns where the boxes of θ = λ \ µ lie: (1) if θ lies in a single column, with number k , then e L n +1 n ( λ, µ ) = ( q N − P j ≥ k m j ( µ ) · (1 − ( − q ) − m k − ( µ ) )(1 − ( − q ) − m k − ( µ ) ) , k > ,q N − P j ≥ m j ( µ ) , k = 1;(2) otherwise, if θ lies in two consecutive columns, with numbers k and k + 1 , then e L n +1 n ( λ, µ ) = ( q N − P j ≥ k m j ( µ ) · ( q − − ( − q ) − m k − ( µ ) ) , k > ,q N − P j ≥ m j ( µ ) · ( q − , k = 1 . In both variants, (1) and (2), the second formula (the one with k = 1) can be viewed as aparticular case of the ﬁrst formula, provided we agree that m ( µ ) = + ∞ and ( − q ) − m ( µ ) := 0.Note also that in variant (1), the right-hand side vanishes if k > m k − ( µ ) ≤

1, whichagrees with the fact that in such a case appending a vertical domino to the k th column of µ isimpossible. Likewise, in variant (2), the right-hand side vanishes if k > m k − ( µ ) = 0,which agrees with the fact that then appending a box to the k th column is impossible. Deﬁnition 8.3.

The branching graph Γ U (2 ∞ ,q ) (resp. Γ U (2 ∞ +1 ,q ) ) is deﬁned by the followingdata:— the set Y E (resp. Y O ) of partitions of even size (resp. odd size) is the set of vertices,and the disjoint union Y E = F n ≥ Y n (resp. Y O = F n ≥ Y n +1 ) deﬁnes a grading on thevertices;— an edge connects µ ∈ Y n and λ ∈ Y n +2 (resp. µ ∈ Y n +1 and λ ∈ Y n +3 ) iﬀ µ րր λ ;the associated edge weight is e L n +1 n ( λ, µ ).This formulation is justiﬁed by the following lemma. Recall that we adopt the deﬁnitionof branching graphs given in Deﬁnition 4.1. Lemma 8.4.

The graded graphs with the vertex sets Y E and Y O and the edges µ րր λ satisfy the conditions listed in Deﬁnition 4.1.Proof. The only condition which is not evident is the following one: for any vertex λ of level n ≥ λ ∈ Y n or λ ∈ Y n +1 , depending on the parity), there should exist avertex µ րր λ .In fact, we can check that there exists a path µ (0) րր · · · րր µ ( n ) = λ in the graph,where the starting point µ (0) is the empty diagram or the one-box diagram, depending onthe parity. To check this, use the well-known fact that an arbitrary Young diagram canbe reduced to its 2 -core (which is either empty or a staircase shape ( m, m − , . . . , (cid:3) The following lemma is proved exactly in the same way as Lemma 4.17.

Lemma 8.5.

There is a bijective correspondence between measures P ∈ P U (2 ∞ ,q )0 (resp., P ∈P U (2 ∞ +1 ,q )0 ) and nonnegative harmonic functions ϕ Γ on the branching graph Γ = Γ U (2 ∞ ,q ) (resp., Γ = Γ U (2 ∞ +1 ,q ) ), uniquely characterized by the property that for any N = 2 n (resp. N = 2 n + 1 ) and any nilpotent matrix X ∈ u ( N, q ) of a given Jordan type µ ∈ Y N , one has P (Cyl N ( X )) = ϕ Γ ( µ ) . (8.2) In the equation (8.2) , Cyl N ( X ) stands for Cyl E N ( X ) if P ∈ P U (2 ∞ ,q )0 , and for Cyl O N ( X ) if P ∈ P U (2 ∞ +1 ,q )0 . Even and odd HL-deformed Young graphs.

Recall the Q -HL functions Q λ (; t ), λ ∈ Y , discussed in Section 4.2. In that section, the parameter t belonged to the interval(0 , t ∈ ( − , Q -HL functions as follows: e Q λ (; t ) := ( − n ( λ ) Q λ (; t ) , λ ∈ Y . Given µ ∈ Y , consider the expansion (cid:0) (1 − t ) p (cid:1) · e Q µ (; t ) = X λ ξ λ/µ ( t ) e Q λ (; t ) . Here, λ ranges over the set of Young diagrams with | λ | = | µ | + 2 and ξ λ/µ ( t ) are certaincoeﬃcients.In the next theorem we compute these coeﬃcients. Let, as usual, m k ( µ ) denotes thenumber of rows in µ of a given length k . We also agree that m ( µ ) = + ∞ and so t m k − ( µ ) = t m k − ( µ ) − = 0 for k = 1. Theorem 8.6.

The coeﬃcient ξ λ/µ ( t ) vanishes unless µ րր λ . Next, suppose µ րր λ andset θ = λ \ µ ; then we have: (1) if θ lies in a single column, with number k , then ξ λ/µ ( t ) = (1 − t m k − ( µ ) )(1 − t m k − ( µ ) − );(2) if θ lies in two consecutive columns, with numbers k and k + 1 , then ξ λ/µ ( t ) = ( − t ) m k ( µ ) (1 + t )(1 − t m k − ( µ ) ) . Here is an immediate corollary.

Corollary 8.7.

Suppose t ∈ ( − , . Then the coeﬃcients ξ λ/µ ( t ) are strictly positive foreach pair of diagrams µ րր λ .Proof of Theorem 8.6. From the deﬁnition of the one-row HL functions Q ( r ) (; t ), it followsthat (1 − t ) p = 2 Q (2) (; t ) − ( Q (1) (; t )) . Next, in the expansion Q ( r ) (; t ) Q µ (; t ) = X λ ψ λ/µ ( t ) Q λ (; t ) , r = 1 , , . . . , the coeﬃcients ψ λ/µ ( t ) vanish unless λ ≻ µ , meaning that λ ⊃ µ and λ \ µ is a horizontalstrip (of length r ), see [16, Ch. III, (5.7 ′ )]. It follows that ξ λ/µ ( t ) = ( − n ( λ ) − n ( µ ) ψ λ/µ ( t ) − X ν : µ ր ν ր λ ψ ν/µ ( t ) ψ λ/ν ( t ) ! . (8.3)Below we use the recipe for computing the coeﬃcients ψ λ/µ ( t ) for λ ≻ µ , explained in [16,Ch. III, (5.8 ′ )]. We examine three possible cases. θ is a vertical domino lying in column k . Then θ is not a horizontal strip, so that ψ λ/µ = 0. Next, there is a single ν situated between µ and λ , and ψ ν/µ ( t ) = 1 − t m k − ( µ ) , ψ λ/ν ( t ) = 1 − t m k − ( µ ) − . Finally, n ( λ ) − n ( µ ) = 2 µ ′ k + 1 is odd. This leads to the formula in (1).2. θ lies in two consecutive columns, with numbers k and k + 1. Then ψ λ/µ = 1 − t m k − ( µ ) .Next, there are two intermediate diagrams ν and we have X ν : µ ր ν ր λ ψ ν/µ ( t ) ψ λ/ν ( t ) = (1 − t m k − ( µ ) )(1 − t m k ( µ )+1 ) + (1 − t m k ( µ ) )(1 − t m k − ( µ ) )Finally, we have n ( λ ) − n ( µ ) = µ ′ k + µ ′ k +1 , which has the same parity as µ ′ k − µ ′ k +1 = m k ( µ ).This leads to the formula in (2).3. θ lies in two columns, with numbers k and ℓ , where ℓ > k + 1. We claim that in thiscase the diﬀerence in (8.3) is equal to 0, so that ξ λ/µ ( t ) = 0. Indeed, we have2 ψ λ/µ ( t ) = 2(1 − t m k − ( µ ) )(1 − t m ℓ − ( µ ) ) . Next, there is two variants for ν , and each of them produces the above expression, withoutthe prefactor 2. Thus, the diﬀerence in (8.3) vanishes.This completes the proof. (cid:3) By virtue of Corollary 8.7 and Lemma 8.4, the following deﬁnition makes sense (cf. Deﬁ-nition 4.7).

Deﬁnition 8.8.

Let t ∈ ( − ,

0) be arbitrary. The even HL-deformed Young graph Y HL E ( t )(resp. odd HL-deformed Young graph Y HL O ( t )) is the branching graph given by:— Y E = F n ≥ Y n (resp. Y O = F n ≥ Y n +1 ) is the graded set of vertices;— an edge connects µ ∈ Y n and λ ∈ Y n +2 (resp. µ ∈ Y n +1 and λ ∈ Y n +3 ) iﬀ µ րր λ ,and then the corresponding edge weight is ξ λ/µ ( t ).8.3. Final result.Theorem 8.9. (i)

The measures P ∈ P U (2 ∞ ,q )0 ( resp., P ∈ P U (2 ∞ +1 ,q )0 ) are in one-to-onecorrespondence with the nonnegative harmonic functions ϕ on the graphs Y HL E ( − q − ) ( resp., Y HL O ( − q − )) .This correspondence P ↔ ϕ is uniquely determined by the property that the mass of anelementary cylinder set Cyl N ( X ) , where N = 2 n is even ( resp., N = 2 n + 1 is odd ) and X ∈ Nil( u ( N, q )) is a nilpotent matrix of Jordan type λ ∈ Y N , is given by P (Cyl N ( X )) = q n ( λ ) − N ( N − / ϕ ( λ ) . Here,

Cyl N ( X ) stands for Cyl E N ( X ) if P ∈ P U (2 ∞ ,q )0 and for Cyl O N ( X ) if P ∈ P U (2 ∞ +1 ,q )0 . (ii) The correspondence P ↔ ϕ establishes aﬃne-isomorphisms of convex cones, P U (2 ∞ ,q )0 ↔ H + ( Y HL E ( − q − )) , P U (2 ∞ +1 ,q )0 ↔ H + ( Y HL O ( − q − )) . In particular, ergodic measures P correspond precisely to extreme harmonic functions ϕ . We need a lemma linking the quantities e L n +1 n ( λ, µ ) computed in Proposition 8.2 with thecoeﬃcients ξ λ/µ ( t ) computed in Theorem 8.6. Recall the function deﬁned in (4.12): f ( λ ) = q n ( λ ) − ( | λ | ) , λ ∈ Y , Lemma 8.10.

Let µ ∈ Y N , λ ∈ Y N +2 , where N = 2 n or N = 2 n + 1 . If µ րր λ , then e L n +1 n ( λ, µ ) = ξ λ/µ ( − q − ) · f ( µ ) f ( λ ) = ξ λ/µ ( − q − ) · q n ( µ ) − n ( λ )+2 N +1 . (8.4) Proof.

Simple direct veriﬁcation. We compare the formulas of Proposition 8.2 with those ofTheorem 8.6. There are two variants that we denoted as (1) and (2). We use the relations n ( λ ) − n ( µ ) = ( µ ′ k + 1 in the case (1) µ ′ k + µ ′ k +1 = 2 µ ′ k − m k ( µ ) in the case (2)and X j ≥ k m j ( µ ) = µ ′ k . (cid:3) Proof of Theorem 8.9. (i) This follows from the chain of bijections P ↔ ϕ Γ ↔ ϕ, where the ﬁrst bijection is given by Lemma 8.5 and the second bijection is given by Lemma8.10, which establishes the similarity of the branching graphs with edge weights e L n +1 n ( λ, µ )and ξ λ/µ ( − q − ).(ii) This claim is a consequence of (i). (cid:3) It is interesting to compare Proposition 4.16 (for the branching graph Y HL ( t ) related to GL ( ∞ , q )) and Theorem 8.9 (for the branching graphs Y HL E ( t ), Y HL O ( t ) related to U (2 ∞ , q ), U (2 ∞ + 1 , q )). The formulas specifying the link between invariant measures and harmonicfunctions on HL-deformed graphs look identical, with the main diﬀerence being that in theformer case, the HL parameter t is specialized to q − , while in the latter case, it is specializedto − q − .It is known that the sign ﬂip q ↔ − q arises in the representation theory of the ﬁniteunitary groups. Namely, the images of the irreducible characters of GL ( n, q ) and U ( n, q ),under appropriate characteristic maps, coincide after the sign ﬂip q ↔ − q ; see [7], [8], [20].This phenomenon is called Ennola’s duality . Our results suggest that a version of Ennola’sduality might exist in our inﬁnite-dimensional setting.It is an open problem to describe explicitly the set of nonnegative harmonic functions onthe graphs Y HL E ( t ) and Y HL O ( t ) with negative t ∈ ( − , Y HL ( t ) with positive t ∈ (0 ,

1) (Problem 1.3 from the introduction).

Remark 8.11.

For the HL-deformed graph Y HL ( t ) (in fact, for more general branchinggraphs), Kerov was able to obtain a list of extreme nonnegative harmonic functions, andconjectured the completeness of this list. A special case of his construction is described indetail in [10, Sect. 4]. The conjecture was proved much later by Matveev [17]; in the caseof our interest, the result is stated in Proposition 4.11. Kerov’s construction is based on thecoalgebra structure of the ring of symmetric functions. It is unclear whether his method canbe adapted to the graphs Y HL E ( t ) and Y HL O ( t ). Examples.

The problem just stated can be reformulated as follows (cf. Remark 4.8).Consider the decomposition Sym = Sym E ⊕ Sym O , where Sym E and Sym O are the linearsubspaces spanned by the homogeneous elements of even and odd degree, respectively. Next,for t ∈ ( − , C HL E ( t ) ⊂ Sym E and C HL O ( t ) ⊂ Sym O be the convex cones spanned bythe functions e Q λ (; t ), where | λ | is assumed to be even or odd, respectively. Then we areinterested in linear functionals Φ on Sym E or Sym O , subject to the following conditions:Φ( p F ) = Φ( F ) for any F ∈ Sym E or F ∈ Sym O ( p -harmonicity); (8.5)Φ is nonnegative on C HL E ( t ) or C HL O ( t ), respectively (positivity) . (8.6)These functionals form convex cones which are in a natural bijective correspondence withthe cones of nonnegative harmonic functions. Adding the extra normalization conditionΦ(1) = 1 if Φ : Sym E → R , or Φ( p ) = 1 if Φ : Sym O → R , we obtain bases of the cones.In this subsection, we construct examples of such functionals for a general value of t ∈ ( − , t = − q − , the functionals give rise to invariant measures, via the relation P (Cyl N ( X )) = q n ( λ ) − N ( N − (1 − q − ) N · Φ( e Q λ (; − q − )) , (8.7)where X ∈ u ( N, q ) is any nilpotent matrix of Jordan type λ .8.4.1. Plancherel-type functionals.

Consider the basis { p ρ : ρ ∈ Y } in Sym formed by theproducts of power-sums. We deﬁne the functionals Φ Planch E : Sym E → R and Φ Planch O :Sym E → R as follows (below n = 0 , , , . . . ):Φ Planch E ( p ρ ) = ( , ρ = (2 n ) , , otherwise; Φ Planch O ( p ρ ) = ( , ρ = (2 n , , , otherwise . Proposition 8.12.

These functionals satisfy the conditions (8.5) – (8.6) .Proof. The p -harmonicity property is clear from the very deﬁnition. Let us check thepositivity. In fact we will prove a stronger claim:Φ Planch E ( e Q λ (; t )) > , λ ∈ Y n ; Φ Planch O ( e Q λ (; t )) > , λ ∈ Y n +1 . (8.8)Below ( · , · ) is the scalar product in Sym depending on the HL parameter t , see [16, Ch.III, Sect. 4]. We use the fact that both { Q λ (; t ) : λ ∈ Y } and { p ρ : ρ ∈ Y } are orthogonalbases.Examine the even case. For λ ∈ Y n , we haveΦ Planch E ( e Q λ (; t )) = ( e Q λ (; t ) , p n )( p n , p n ) . By [16, Ch. III, (4.11)], ( p n , p n ) = (1 − t ) − n · n n ! > . Next, let ξ λ/ ∅ stand for the coeﬃcient of e Q λ (; t ) in the expansion (cid:0) (1 − t ) p (cid:1) n = X ν ∈ Y n ξ ν/ ∅ ( t ) e Q ν (; t ) . Then we have( e Q λ (; t ) , p n ) = ξ λ/ ∅ ( t ) b λ ( t )(1 − t ) n , b λ ( t ) := ( Q λ (; t ) , Q λ (; t )) = ( e Q λ (; t ) , e Q λ (; t )) . A formula for the quantity b λ ( t ) was previously displayed in (4.2); it clearly shows that b λ ( t ) >

0, if t ∈ ( − , ξ λ/ ∅ ( t ) >

0. But this coeﬃcient equals thesum of the weights of all paths ∅ րր · · · րր λ joining λ to the root ∅ in the graph Y HL E ,where the weight of a path is equal to the product of its edge weights. By Lemma 8.4, theset of these paths is nonempty, which entails the desired inequality.In the odd case the argument is similar. In this case the root of Y HL O is the one-boxdiagram. (cid:3) Functionals connected with the principal specialization.

Consider the specializationsSym → R deﬁned byΦ m ( F ) := F ( c m , c m t, . . . , c m t m − ) , m = 1 , , . . . ; Φ ∞ ( F ) := F ( c ∞ , c ∞ t, c ∞ t , . . . ) , (8.9)where the positive constants c , c , . . . , c ∞ are chosen so that the value at F = p equals 1: c m = s − t − t m , c ∞ = √ − t . We are interested in the restrictions of Φ m and Φ ∞ to Sym E and to Sym O . Proposition 8.13.

The functionals Φ m , Φ ∞ satisfy the conditions (8.5) – (8.6) , both for theeven and odd cases.Proof. The p -harmonicity property is evident, and the positivity property follows from thefact that (see [16, Ch. III, Sect. 4, ex. 3])Φ m ( Q λ (; t )) = c | λ | m t n ( λ ) ℓ ( λ ) Q i =1 (1 − t m − i +1 ) , ℓ ( λ ) ≤ m, , ℓ ( λ ) > m ; Φ ∞ ( Q λ (; t )) = c | λ |∞ t n ( λ ) . Since ( − n ( λ ) t n ( λ ) > t , we obtain that Φ m ( e Q λ (; t )) is strictly positive for ℓ ( λ ) ≤ m and is vanishing otherwise, whereas Φ ∞ ( e Q λ (; t )) > λ . (cid:3) Remark 8.14.

The invariant measure P corresponding to Φ ∞ is the restriction of the(properly normalized) Haar measure to the subset of pronilpotent matrices, both in evenand odd cases (cf. Example 4.21). In fact, the corresponding measure P is determined bythe relations (see (8.7)): P (Cyl N ( X )) = q − N ( N − , X ∈ u ( N, q ) , N = 0 , , · · · . Evidently, P is the restriction of a Haar measure because P (Cyl N ( X )) does not depend onthe Jordan type of X .8.4.3. One more family of functionals.

The functional Φ from (8.9) is the specialization F F ( √ t , t √ t ). The following functionals form a one-parameter deformation of thisspecialization.Let a , a be real numbers such that a > > a , a + a = 1, and a > | a | . LetΦ a ,a : Sym → R be the specialization F F ( a , a ).6 CESAR CUENCA AND GRIGORI OLSHANSKI

The functional Φ from (8.9) is the specialization F F ( √ t , t √ t ). The following functionals form a one-parameter deformation of thisspecialization.Let a , a be real numbers such that a > > a , a + a = 1, and a > | a | . LetΦ a ,a : Sym → R be the specialization F F ( a , a ).6 CESAR CUENCA AND GRIGORI OLSHANSKI Proposition 8.15.

The functionals Φ a ,a satisfy the conditions (8.5) – (8.6) , both for theeven and odd cases.Proof. The p -harmonicity property holds, because Φ a ,a ( p ) = a + a = 1. Next, we haveΦ a ,a ( e Q λ (; t )) = 0 if ℓ ( λ ) >

2. We are going to show that Φ a ,a ( e Q λ (; t )) ≥ λ with ℓ ( λ ) ≤ λ = ∅ . Suppose ℓ ( λ ) = 1, so that λ = ( m ) with m = 1 , , . . . . Then n ( λ ) = 0, so that we have to check that Q ( m ) ( a , a ; t ) >

0. From [16, Ch. III, (2.9)], Q ( m ) ( a , a ; t ) = (1 − t ) · (cid:18) a m · a − ta a − a + a m · a − ta a − a (cid:19) (8.10)Since 1 − t > a − a >

0, the desired inequality reduces to a m ( a − ta ) − a m ( a − ta ) ? > . (8.11)Since the left hand side of (8.11) is homogeneous on a , a , we can assume that a = − a = a >

1. Then the inequality turns into a m +1 + ( − m ? > − t ( a m + ( − m a ) . Since t ∈ ( − ,

0) and a >

1, it suﬃces to prove that a m +1 + ( − m ? ≥ a m + ( − m a, but this follows from the evident inequality a m ( a − ≥ ( − m ( a − . It remains to examine the case ℓ ( λ ) = 2. Then λ = ( m, n ) for some integers m ≥ n ≥ n ( λ ) = n ; using this we obtain e Q ( m,n ) ( a , a ; t ) = ( − n Q ( m,n ) ( a , a ; t ) = ( − n ( a a ) n Q ( m − n ) ( a , a ; t )= | a a | n Q ( m − n ) ( a , a ; t ) > , where the last inequality holds by the case already considered. (cid:3) Part Appendices

Appendix A. Proof of Propositions 4.13 and 8.2

A.1.

Two lemmas.

Let V be a ﬁnite-dimensional vector space. Given an operator A on V , we denote by Ran A its range and set rk A := dim(Ran A ).The following lemma shows how to ﬁnd the Jordan type of a nilpotent operator. Lemma A.1.

Let X be a nilpotent operator on V , µ ∈ Y be its Jordan type, and µ ′ be thetransposed diagram. Then rk X i − − rk X i = µ ′ i , for all i ≥ . As a consequence, rk X k = X j ≥ k +1 µ ′ j , for all k ≥ . Proof.

Easy exercise using the fact that if A is a p × p nilpotent matrix of Jordan type ( p ),then rk A k = max( p − k, (cid:3) Let X be as in Lemma A.1. We assign to it the nested sequence of subspaces { } = V ⊆ V ⊆ V ⊆ V ⊆ . . . , where V k := { v ∈ V : X k − v ∈ Ran X k } , k = 1 , , . . . (A.1)(the fact that V k ⊆ V k +1 follows directly from the very deﬁnition). Lemma A.2.

Let X be as above and µ be its Jordan type. We have dim V k = dim V − X j ≥ k m j , k ≥ , where, as before, m j = m j ( µ ) := { i : µ i = j } . In particular, we see that V k = V for k ≥ µ + 1. Proof.

Examine ﬁrst the special case when µ = ( p ), where p := dim V . Pick a basis v , . . . , v p of V such that Xv i = v i − for i = 1 , . . . , p , where v := 0. Then V k = ( span { v , . . . , v p − } , ≤ k ≤ p,V, k ≥ p + 1 . More generally, if µ is arbitrary, we pick a basis { v ij } indexed by the boxes ( i, j ) ∈ µ suchthat Xv ij = v i,j − with the understanding that v i := 0 (see the proof of Proposition 6.1).Then V k is the linear span of the basis vectors v ij such that ( i, j ) satisﬁes one of the followingtwo conditions: • µ i ≤ k −

1, or • µ i ≥ k and 1 ≤ j ≤ µ i − i, j ) equals | µ | − P j ≥ k m j , which completes the proof. (cid:3) A.2.

Proof of Proposition 4.13.

We set V = F n and ﬁx a nilpotent operator X on V ,which we identify with the corresponding n × n matrix. Let µ ∈ Y n denote the Jordan typeof X . Next, given x ∈ V , we consider the matrix Y := (cid:20) X x (cid:21) and denote by λ its Jordan type. Our task is to compute the number of vectors x ∈ V leading to a given λ ∈ Y n +1 ; let us denote this quantity by L ( λ, µ ) (instead of the moredetailed notation L n +1 n ( λ, µ ), as in the original formulation of the proposition).Set ǫ i := rk Y i − rk X i , i = 0 , , , . . . . By Lemma A.1, λ ′ i − µ ′ i = ǫ i − − ǫ i , i = 1 , , , . . . . (A.2)We have ǫ = 1, because Y and X are the identity matrices of size n + 1 and n ,respectively. Next, Y i = (cid:20) X i X i − x (cid:21) , i ≥ . For a matrix A , let colsp( A ) denote the space of vectors spanned by the columns of A .We have ǫ i = ( , X i − x ∈ colsp( X i ) , , X i − x / ∈ colsp( X i ) , i = 1 , , . . . . On the other hand, the condition X i − x ∈ colsp( X i ) just means x ∈ V i . Therefore theabove formula can be rewritten as ǫ i = ( , x ∈ V i , , x / ∈ V i , i = 1 , , . . . . Given x ∈ V , let k = k ( x ) be the smallest positive integer such that x ∈ V k . Then weobtain ǫ i = ( , ≤ i < k, , i ≥ k. From this and (A.2) (and taking into account the equality ǫ = 1) we obtain λ ′ i − µ ′ i = ( , i = k, , i = k. In words: λ is obtained from µ by adding a box to the k -th column.For such λ , we have, by the deﬁnition of k = k ( x ), L ( λ, µ ) = ( q dim V k − q dim V k − , k ≥ ,q dim V , k = 1 . Finally, applying Lemma A.2, we obtain the desired expression (4.10).A.3.

One more lemma.

We keep to the deﬁnitions and notation introduced in SectionA.1. Assume additionally that the base ﬁeld is e F , the e F -vector space V is equipped with anondegenerate sesquilinear Hermitian form τ , and X is a τ -Hermitian nilpotent operator on V . Lemma A.3.

Let k ≥ . Given two vectors x, y ∈ V k , pick a vector x ′ ∈ V such that X k − x = X k x ′ ( which is possible by the very deﬁnition of V k ) . (i) The quantity τ ( X k − x ′ , y ) does not depend on the choice of x ′ . (ii) The map τ k : V k × V k → e F deﬁned by τ k ( x, y ) := τ ( X k − x, y ) − τ ( X k − x ′ , y ) is a Hermitian form. (iii) The radical

Rad τ k of this form is equal to V k − .Proof. Consider the model of X described in the proof of Proposition 6.1. Using it one canimmediately reduce the general situation to the particular case when X consists of a singleJordan block. Then we may suppose that V has a basis v , . . . v p such that Xv i = v i − (where v := 0) and τ ( v i , v j ) = δ i + j,p +1 . Below we write x = ( x , . . . , x p ) meaning that x = P x i v i .A similar notation is used for x ′ and y .We examine separately three possible cases: k ≥ p + 2, k = p + 1, and 2 ≤ k ≤ p . • k ≥ p + 2. Then V k = V k − = V . On the other hand, X k − x = X k − x ′ = 0 for all x, x ′ ∈ V . So all claims hold for trivial reasons. • k = p + 1. Then V k = V p +1 = V and V k − = V p consists of the vectors with lastcoordinate 0. Next, the condition X k − x = X k x ′ amounts to X p x = X p +1 x ′ , which holdsfor any choice of x ′ because both sides equal 0. Thus, x ′ may be arbitrary. On the otherhand, X k − x ′ = X p x ′ = 0, so τ k ( x, y ) does not depend on x ′ and we have τ k ( x, y ) = τ ( X p − x, y ) = x p ¯ y p . Again, all claims hold true. • ≤ k ≤ p . Then V k = V k − , which means that τ k must be identically equal to 0. Letus take any x, y ∈ V k and verify that τ k ( x, y ) = 0. The space V k is the subspace of vectorswith the last coordinates equal to 0. Thus, x p = y p = 0. Next, given x , we may take x ′ = (0 , x , . . . , x p − ). With this choice we have x = Xx ′ and hence τ k ( x, y ) = 0, as desired.Any other possible choice for x ′ consists in adding a vector u ∈ ker X k . For such a vectorwe have u k +1 = · · · = u p = 0. Then we have τ ( X k − u, y ) = u k ¯ y p = 0 , because y p = 0. Thus, we still get τ k ( x, y ) = 0.This completes the proof. (cid:3) A.4.

Proof of Proposition 8.2.

It is convenient to switch from skew-Hermitian matrices(as in the original formulation of the proposition) to Hermitian matrices. This is done simplyfrom the relation Herm(

N, q ) = ǫ · u ( N, q ), see Section 6.Let V = e F N and τ : V × V → e F be the Hermitian form given by the matrix W = W N .We ﬁx a τ -Hermitian nilpotent operator X on V , which we identify with the corresponding N × N matrix. Let µ ∈ Y N denote the Jordan type of X . Next, given x ∈ V and z ∈ F , weform the matrix Y := x ∗ W z X x and denote by λ its Jordan type. Our task is to compute the number of pairs ( x, z ) ∈ V × F leading to a given λ ∈ Y N +2 ; let us denote this quantity by e L ( λ, µ ) (instead of e L n +1 n ( λ, µ ),as in the original formulation of the proposition).We have Y i = x ∗ W X i − z i X i X i − x , i ≥ , where z i = ( z, i = 1 ,x ∗ W X i − x, i ≥ . (A.3)Note that W X m = ( X ∗ ) m W, m ≥ , (A.4)because X is τ -Hermitian.As before, we set ǫ i := rk Y i − rk X i , i ≥ , and we still have λ ′ i − µ ′ i = ǫ i − − ǫ i , i ≥ . (A.5) Obviously, ǫ = 2 (not 1 as before!) andrk Y i = rk Y [ i ] , Y [ i ] := (cid:20) x ∗ W X i − z i X i X i − x (cid:21) , i ≥ . Thus, ǫ i := rk Y [ i ] − rk X i , i ≥ . Recall the notation colsp( A ) for the space of column vectors spanned by the columns ofa given matrix A . Likewise, let rowsp( A ) denote the space of row vectors spanned by therows of A .Observe that X i − x ∈ colsp( X i ) ⇔ x ∗ W X i − ∈ rowsp( X i ) , because X is τ -Hermitian. Recall that this condition is also equivalent to x ∈ V i , see (A.1).Given i ≥

1, we examine two possible cases depending on whether x lies in V i or not. • If x / ∈ V i , then X i − x / ∈ colsp( X i ) and x ∗ W X i − / ∈ rowsp( X i ), which entails that ǫ i = 2. • Suppose now that x ∈ V i . By the deﬁnition of V i , there exists a vector x ′ ∈ V such that X i x ′ = X i − x . Then we may kill the lower-right block of the matrix Y [ i ] by subtracting fromthe last column the linear combination of the ﬁrst N columns with the coeﬃcients equal tothe coordinates of x ′ . Next, we can also kill the upper-left block in a similar way. This leadsus to the matrix e Y [ i ] := (cid:20) z i − x ∗ W X i − x ′ X i (cid:21) = (cid:20) − ( x ′ ) ∗ W (cid:21) Y [ i ] (cid:20) − x ′ (cid:21) , which has the same rank as Y [ i ] (the second equality is veriﬁed with the use of (A.4)).This gives us: ǫ i = ( , z i − x ∗ W X i − x ′ = 0 , , z i − x ∗ W X i − x ′ = 0 . (A.6)We also note that, by virtue of (A.3) and Lemma A.3, z i − x ∗ W X i − x ′ = x ∗ W X i − x − x ∗ W X i − x ′ = τ i ( x, x ) , i ≥ . Now we are in a position to describe the possible form of λ and compute the desiredquantity e L ( λ, µ ).As before, we denote by k = k ( x ) the least positive integer such that x ∈ V k . Since V i = V for all i ≥ µ + 1, we have 1 ≤ k ≤ µ i + 1. We examine separately two cases linked to thealternative in (A.6). Then each of them is subdivided into two cases depending on whether k ≥ k = 1.Case (1A): k ≥ τ k ( x, x ) = 0. We claim that in this case ǫ i = ( , ≤ i ≤ k − , , i ≥ k. Indeed, we always have ǫ = 2. If 1 ≤ i ≤ k −

1, then x / ∈ V i , whence ǫ i = 2. For i = k we have ǫ k = 0, because x ∈ V k \ V k − and τ k ( x, x ) = 0 by the assumption. For i ≥ k + 1the same result holds because we still have x ∈ V i and τ i ( x, x ) = 0, where the latter equalityholds for the reason that x ∈ V i − and V i − is the radical of τ i (Lemma A.3). Case (1B): k = 1 and z − x ∗ W x ′ = 0. (Here we assume that x ′ is chosen in advance andis ﬁxed.) We claim that in this case ǫ i = ( , i = 0 , , i ≥ . The proof is as the same as in the previous case, the only diﬀerence is that the role of thequantity τ ( x, x ) (which is not deﬁned) is played by z − x ∗ W x ′ = z − x ∗ W x ′ .Case (2A): k ≥ τ k ( x, x ) = 0. We claim that in this case ǫ i = , ≤ i ≤ k − , , i = k, , i ≥ k + 1 . Here we argue again as in Case (1A); the only diﬀerence arises for i = k because now τ k ( x, x ) = 0.Case (2B): k = 1 and z − x ∗ W x ′ = 0. (Again, x ′ is ﬁxed in advance.) We claim that inthis case ǫ i = , i = 0 , , i = 1 , , i ≥ . Indeed, this is checked as in Case (1B).From (A.5) and the above expressions for ǫ i it is seen that in Case (1) (that is, (1A) and(1B)), λ is obtained from µ by appending a vertical domino to the k th column, while in Case(2) (that is, (2A) and (2B)), one box is appended to the k th column and another box isappended to the ( k + 1)th column. Thus, in both cases, µ րր λ , as claimed in Proposition8.2.We proceed to the calculation of e L ( λ, µ ). We need a notation: Deﬁnition A.4.

Let E be a ﬁnite-dimensional vector space over e F and κ : E × E → e F anondegenerate sesquilinear Hermitian form. Then we set c ( E ) := { x ∈ E : κ ( x, x ) = 0 } , c ( E ) := { x ∈ E : κ ( x, x ) = 0 } . (A.7)Observe that the choice of κ does not aﬀect the deﬁnition of c ( E ) and c ( E ), because allforms κ on the vector space E are all equivalent.Note that c ( E ) + c ( E ) = q E . We claim that the following formulas hold:Case (1A): e L ( λ, µ ) = q ( c ( V k /V k − ) − q V k − , k = 2 , , . . . , Case (1B): e L ( λ, µ ) = q V , Case (2A): e L ( λ, µ ) = qc ( V k /V k − ) q V k − , k = 2 , , . . . , Case (2B): e L ( λ, µ ) = q V ( q − . From (A.7) it follows that the sum of these expressions equals q V +1 , as it should be(because this is the the total number of pairs ( x, z ) ∈ V × F ).These formulas are derived directly from the deﬁnition of the cases. For instance, let uscomment on Case (1A). Here z ∈ F may be arbitrary, whence the factor q . Next, we have tocount the number of vectors x ∈ V k \ V k − with τ k ( x, x ) = 0. Recall that V k − is the radicalof the form τ k , so that we may treat τ k as a nondegenerate form on V k /V k − . The numberof vectors x with the required conditions is equal to the number of pairs ( u, v ), where u is a nonzero vector in the quotient space V k /V k − with τ k ( u, u ) = 0, and v is any vector in V k − .The number of possible vectors u is c ( V k /V k − ) −

1, whereas the number of possible vectors v is q V k − – the cardinality of V k − . The veriﬁcation of the other formulas is similar.To complete the computation we need the following explicit expressions for c ( E ) and c ( E ) (see Lemma A.5 below): c ( E ) = q m − + ( − m q m − ( q − , c ( E ) = q m − c ( E ) , m := dim E. (A.8)Finally, we need explicit expressions for the dimensions entering our formulas, and theyare aﬀorded by Lemma A.2:dim( V k /V k − ) = m k − , dim V k − = N − X j ≥ k − m j , dim V = N − X j ≥ m j . (A.9)Substituting (A.8) and (A.9) into the expressions for e L ( λ, µ ) we get, after simple trans-formations, the formulas of Proposition 8.2.It remains to check (A.8). Lemma A.5.

The quantities c ( E ) and c ( E ) deﬁned by (A.7) are given by (A.8) .Proof. Write c ( m ) and c ( m ) instead of c ( E ) and c ( E ). The equality c ( m ) = q m − c ( m )is evident. Next, c ( m ) equals the number of solutions in e F m of the equation x x + · · · + x m x m = 0 . Clearly, c (1) = 1, which agrees with (A.8). Next, for m ≥ a := x x + · · · + x m − x m − and observe that the equation x m x m = − a has a single solution x m = 0 for a = 0, and q + 1solutions for every a ∈ F \ { } . This entails the recurrence relation c ( m ) = c ( m −

1) + c ( m − q + 1) , m = 2 , , . . . . On the other hand, (A.8) satisﬁes this recurrence, which proves the lemma. (cid:3)

This completes the proof of Proposition 8.2.

Appendix B. Generalized spherical representations

The contents of this appendix are not used in the body of the paper, so we omit detailedproofs. Our purpose here is to explain a representation-theoretic meaning of coadjoint-invariant Radon measures.Let V be a locally compact Abelian (LCA) group, and b V its Pontryagin dual (the groupof characters). Next, let K be a compact group acting on V by automorphisms. Then italso acts by automorphisms on b V , and we set G := K ⋉ b V (the semidirect product). By a spherical representation of ( G , K ) we mean a pair ( T, ξ ), where T is a unitary representation of the group G on a Hilbert space H = H ( T ) and ξ ∈ H is a distinguished cyclic K -invariantvector. A well-known result due to Mackey says that there is a one-to-one correspondence P ↔ ( T, ξ ) between ﬁnite K -invariant Borel measures P on V and (equivalence classes of)spherical representations ( T, ξ ) of ( G , K ).This correspondence looks as follows. Given P , we form the Hilbert space H := L ( V, P ).The distinguished vector ξ is the constant function equal to 1. The representation T isdeﬁned by setting for η ∈ H and v ∈ V ( T ( k ) η )( v ) := η ( k − · v ) , k ∈ K , (B.1)and ( T ( b v ) η )( v ) := χ ( b v, v ) , b v ∈ b V , (B.2)where χ denotes the canonical pairing between b V and V (a bicharacter of b V × V ). Then( T, ξ ) is a spherical representation, and every spherical representation is obtained in this wayfrom a (unique) measure P .We are going to extend this correspondence to the case when K is no longer compact andthe measures P are allowed to be inﬁnite. On the other hand, we impose restrictions on V .We need a few deﬁnitions. By a lattice in a LCA group V we mean an open compactsubgroup; we denote by Lat( V ) the set of all lattices in V . Next, if X is a closed subgroupof a LCA group V , then we will denote by X ⊥ the subgroup of b V formed by the charactersthat identically equal 1 on X . Note that if X ∈ Lat( V ), then X ⊥ ∈ Lat( b V ).We are interested in LCA groups V satisfying the following condition:(*) There exists a two-sided inﬁnite sequence of nested lattices { X n : n ∈ Z } such that · · · ⊃ X n ⊃ X n +1 ⊃ · · · , \ X n = { } , [ X n = V. Note that if V satisﬁes (*), then so does b V ; for the corresponding chain of lattices, one cantake Y n := X ⊥− n .In what follows, we assume that V satisﬁes (*). A basic example is the additive group ofa local non-Archimedean ﬁeld F (( z )), where F is a ﬁnite ﬁeld.We denote by S ( V ) the space of Schwartz–Bruhat functions — these are the complexvalued functions on V that are compactly supported and locally constant. Let also S [ b V ] bethe space of complex measures on V which are absolutely continuous with respect to theHaar measure and whose densities are Schwartz–Bruhat functions. Both S ( V ) and S [ b V ] arecommutative ∗ -algebras, with respect to pointwise multiplication and convolution product,respectively.The Fourier transform establishes an algebra isomorphism between S ( V ) and S [ b V ]. Fromthis it is easy to obtain a version of Bochner’s theorem: Proposition B.1.

The Fourier transform establishes a bijection P ↔ ϕ between Radonmeasures P on V and positive functionals ϕ on the ∗ -algebra S [ b V ] . Let us emphasize that it is the Radon condition that allows one to deﬁne the Fouriertransform of P .Suppose now that T is a continuous unitary representation of the group b V on a Hilbertspace H . We are going to deﬁne the Gelfand triple H ∞ ⊂ H ⊂ H ∞ . For X ∈ Lat( b V ), let H X denote the subspace of X -invariant vectors in H . By deﬁnition, H ∞ := [ X ∈ Lat( b V ) H X (the subspace of smooth vectors ). It is an inductive limit space, H ∞ = lim −→ H X , with respectto natural embeddings H X → H Y corresponding to pairs of lattices X ⊃ Y . We equip H ∞ with the inductive limit topology, where each H X is endowed with its usual norm topology.Next, H ∞ is deﬁned as the space of continuous antilinear functionals on H ∞ ; it is nothingelse than the projective limit lim ←− H X , where, for each pair X ⊃ Y , the arrow H X ← H Y isthe natural projection. Elements of H ∞ are called generalized vectors .The representation T gives rise to a ∗ -representation of the algebra S [ b V ] on H , whichpreserves the subspace H ∞ ⊂ H and extends to the space H ∞ . Furthermore, if ξ ∈ H ∞ and f ∈ S [ b V ], then T ( f ) ξ ∈ H ∞ . A generalized vector ξ is said to be cyclic if the subspace { T ( f ) ξ : f ∈ S [ b V ] } is dense in H . Proposition B.2.

There exists a natural bijective correspondence P ↔ ( T, ξ ) between Radonmeasures P on V and (equivalence classes of ) pairs ( T, ξ ) , where T is a unitary representa-tion of b V and ξ is a distinguished generalized cyclic vector.Sketch of proof. Given P , the corresponding pair ( T, ξ ) is constructed as follows. The Hilbertspace H is L ( V, P ). The operator T ( b v ) on H is deﬁned by (B.2). The subspace H ∞ ⊂ H consists of compactly supported L -functions. The space H ∞ ⊃ H is formed by the locally- L -functions. The distinguished vector ξ is the constant function 1.Conversely, given ( T, ξ ), the matrix element ϕ ( b v ) := ( T ( b v ) ξ, ξ )is well deﬁned, and it is a positive functional on S [ b V ]. Then we use the correspondnce ϕ P from Proposition B.1. (cid:3) Suppose now that there is a group K acting on V (and hence on b V ) by automorphisms.Then we may form the semidirect product G := K ⋉ b V . If T is a unitary representation of G on a Hilbert space H , then the action of K preserves the Gelfand triple associated withthe restriction of T to b V . By a generalized spherical representation of ( G , K ) we mean a pair( T, ξ ), where T is as above and ξ ∈ H ∞ is a generalized K -invariant cyclic vector ξ .Denote by P ( V ) K the set of K -invariant Radom measures on V . Given P ∈ P ( V ) K , weassign to it a generalized spherical representation of the group G by formulas (B.1) and (B.2);the distinguished vector ξ being, as before, the constant function 1. Proposition B.3.

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