Intersections of Deligne--Lusztig varieties and Springer fibres
aa r X i v : . [ m a t h . R T ] J a n INTERSECTIONS OF DELIGNE–LUSZTIG VARIETIES AND SPRINGERFIBRES
ZHE CHEN
Abstract.
In this paper we study intersections of Deligne–Lusztig varieties and Springerfibres in type A over finite fields. In particular, we prove a direct geometric relation betweenthe two varieties: For any rational unipotent element, the Springer fibre cut out a uniquecomponent of a specific Deligne–Lusztig variety; moreover, this component forms an opendense subset of a component of the Springer fibre. This combines several constructions witha combinatorial flavour (like Weyr normal forms, Robinson–Schensted correspondence, andSpaltenstein’s and Steinberg’s labellings). Contents
1. Introduction 12. Deligne–Lusztig varieties and Springer fibres 23. Some preliminaries 34. Distribution of components 65. Examples and remarks 10References 151.
Introduction
Let G be a connected reductive group over a finite field F q , and let F be the geometricFrobenius endomorphism on G := G × F q F q . Fix an F -stable Borel subgroup B ⊆ G andan F -stable maximal torus T ⊆ B . There are two important classes of varieties lying in theflag variety G/B , namely, Deligne–Lusztig varieties and Springer fibres.Deligne–Lusztig varieties X w (see Definition 2.1) are parametrised by w ∈ W ( T ) := N ( T ) /T , and their ℓ -adic cohomology (with coefficients in suitable local systems) affordsall the irreducible representations of G F = G ( F q ). Meanwhile, Springer fibres B u (see Defi-nition 2.4) are parametrised by the unipotent elements u ∈ G , and their ℓ -adic cohomologyaffords all the irreducible representations of W ( T ). Since their births in the seminal works[DL76] and [Spr76], respectively, these two families of varieties play crucial roles in the studyof representations of Lie type groups and Weyl groups. In this paper we give a study of theirrelations in the case of type A .In the remaining part of this paper we take G to be GL n , and let B be the standard upperBorel subgroup and T the diagonal maximal torus.Indeed recently we found that, for a specific unipotent u , the intersection B u,w := X w ∩ B u appears in the study of smooth representations of the profinite group GL d ( F q [[ π ]]), where is a divisor of n (see [Che20b]). This serves the initial motivation for our attention onthe relations between these two varieties. We also remark that a similar theme has beenconsidered over C in [Tym06], in which case Hessenberg varieties and Schubert cells took theroles of Springer fibres and Deligne–Lusztig varieties. In any case, providing the facts thatboth X w and B u are vital in geometric representation theory, and that they share the sameambient space G/B , it is very interesting and natural to seek their geometric interactions.We found the following surprising simple relation between the components: (See Theo-rem 4.1 for the formal statement.)(a) Any Springer fibre at a rational unipotent element of G has a component containinga component of a specific Deligne–Lusztig variety as a dense open subset; moreover,this Deligne–Lusztig component forms the whole intersection.(b) Conversely, every component of a Deligne–Lusztig variety at an involution of a specificshape is a dense open subset of a component of some Springer fibre.In Section 2 we make some preparations on Deligne–Lusztig varieties and Springer fibres.In Section 3 we give a brief recall of some ingredients used in the proof of Theorem 4.1, likeWeyr normal forms, Robinson–Schensted correspondence, and Spaltenstein’s and Steinberg’sdescriptions of Springer components.In Section 4 we present the proof of Theorem 4.1, which is a careful combination of theabove constructions. We also derive a geometric proof of a classical dimension formula ofunipotent centralisers (see Corollary 4.2).In Section 5 we give a few remarks, including three illustrating examples (one concerningthe boundary of the theorem, one concerning a uniqueness property, and one concerning anopposite phenomenon), and a short discussion on the representations associated with B u,w .Throughout this paper: We use the convention notation g h = h − g = h − gh for elements g, h in an algebraic group; all varieties are assumed to be reduced; by a component we alwaysmean an irreducible component. Acknowledgement.
The author thanks George Lusztig and Alexander Stasinski for helpfulcomments and suggestions, and thanks Guangyi Yue for a helpful communication. During thepreparation of this work the author is partially supported by the NSFC funding no.12001351.2.
Deligne–Lusztig varieties and Springer fibres
In this section we recall some basics of Deligne–Lusztig varieties and Springer fibres. Thedetails can be found in [DL76], [Sho88], [Ste88], [Car93].
Definition 2.1.
Let w ∈ W ( T ) ∼ = S n . Then the Deligne–Lusztig variety at w is X w := L − ( BwB ) /B, where L : G → G is the Lang isogeny given by g g − F ( g ).Viewing G/B as the variety of complete flags, one can describe X w in the following way. Definition 2.2.
Let F : V ⊆ V ⊆ ... ⊆ V n and F ′ : V ′ ⊆ V ′ ⊆ ... ⊆ V ′ n e two complete flags of V := ( F q ) n , with dim V i = dim V ′ i = i . We say that F and F ′ arein relative position w ∈ W ( T ) = S n , ifdim V i ∩ V ′ j = { , ..., i } ∩ { w (1) , ..., w ( j ) } )for any i, j ∈ { , ..., n } . (*) Let G = GL n be viewed as the automorphism group of V . Then X w is the varietyconsisting of the complete flags F : V ⊆ V ⊆ ... ⊆ V n such that F and F F are inrelative position w , namelydim V i ∩ F V j = { , ..., i } ∩ { w (1) , ..., w ( j ) } )for any i, j ∈ { , ..., n } . Proposition 2.3 (Deligne–Lusztig) . The variety X w is a smooth locally closed subvarietyof G/B of pure dimension l ( w ) , where l ( w ) denotes the length of w .Proof. See [DL76, Page 107]. (cid:3)
Definition 2.4.
Let u ∈ G be a unipotent element. Then the Springer fibre at u is B u := { gB ∈ G/B | u g ∈ B } . Using the term of flags, B u ⊆ G/B can be viewed as the closed subvariety consisting ofcomplete flags F : V ⊆ V ⊆ ... ⊆ V n stabilised by u (namely, uV i = V i for all i ).Unlike X w , usually B u is singular, but one still has: Proposition 2.5 (Spaltenstein, Steinberg) . The variety B u is of pure dimension v G − dim C ( u ) , where v G denotes the number of positive roots and C ( u ) denotes the conjugacyclass of u .Proof. See e.g. [Sho88, 1.2]. (cid:3)
In this paper we will very often take the viewpoint that elements in X w and B u are flags.We put B u,w := B u ∩ X w = B u × G/B X w .3. Some preliminaries
In this section we recall some ingredients needed in the proof of Theorem 4.1. We first fixthe notation that will be used throughout this paper:
Notation 3.1.
Let u ∈ G be a unipotent element. Then • J ( u ) = diag { J , ..., J d } is the standard Jordan normal form of u , where the J i ’s arethe Jordan blocks (with non-increasing sizes). • r i is the size of the Jordan block J i (that is, J i is an r i × r i -matrix); in particular r i ≥ r i +1 . • λ ( u ) is the Young diagram associated with J ( u ), that is, a Young diagram whose i -throw has r i boxes. • c i is the number of boxes in the i -th column of λ ( u ). For convenience, we also put c = 0. Note that there are totally r columns and c = d rows.One ingredient we would need is the so-called Weyr normal form W ( u ), which is a “dual”of the Jordan normal form J ( u ): efinition 3.2. For a unipotent u ∈ GL n ( F q ), the matrix W ( u ) is blocked upper triangular,and is characterised by the following rules:(i) The i -th diagonal block is the c i × c i identity matrix.(ii) The block just right to the i -th diagonal block is of the form ( I ), where I denotesthe c i +1 × c i +1 -identity matrix, and 0 denotes the zero matrix of a suitable size.(iii) All other blocks are zero.So W ( u ) is a blocked matrix of the shape I ( I ) 0 0 ... I ( I ) 0 ...
00 0 I ( I ) ... ... ... ... ... ... ... ... I ( I )0 0 ... I , where the I ’s denote some identity matrices of possibly different sizes and the 0’s denotesome zero matrices of possibly different sizes.Although Weyr normal forms and Jordan normal forms were both discovered in the secondhalf of the 19th century, the Weyr form appears to be much lesser known; a comprehensivereference on these normal forms is [OCV11]. Note that recently there has been a (verydifferent) application of Weyr normal forms in the representation theory of Lie type groupsover local rings; see [Sta21]. Remark 3.3.
One of the main features we need from Weyr form (instead of Jordan form)is that the block sizes are with respect to the columns of Young diagrams, which allows oneto combine the other elements in the proof of Theorem 4.1 in a natural way.
Proposition 3.4.
The unipotent elements u and W ( u ) are in the same conjugacy class.Proof. See e.g. [OCV11, 2.2.2]. (cid:3)
Another ingredient we need is the Robinson–Schensted correspondence. Recall that a verybasic property of finite group representation theory is the identity P ρ (dim ρ ) = H , where H is a finite group and ρ suns over the irreducible representations. If we take H = S n , thenthis identity can be re-written as X λ ( T ( λ )) = S n , where λ runs over the Young diagrams of n boxes, and T ( λ ) denotes the set of standard λ -tableaux (we use the convention that the numbers in a standard tableaux go increasinglyfrom left to right and from up to down). The Robinson–Schensted correspondence, whichwas later generalised by Knuth to a more general situation, gives a combinatorial explanationof this identity. Proposition 3.5 (Robinson–Schensted correspondence) . There is an explicit computablebijection between the sets w ( − , − ) : G λ T ( λ ) × T ( λ ) −→ S n , atisfying the property w ( P, Q ) = w ( Q, P ) − , given by the following algorithm: (i) Take ( P, Q ) ∈ T ( λ ) × T ( λ ) . If n is in the ( i, j ) -th box of Q , then we remove this boxfrom Q , and denote the new tableau by Q ′ . (ii) Suppose the number in the ( i, j ) -th box of P is n ′ . We remove this box from P andmove n ′ up by one row to replace the largest number smaller than n ′ . (iii) Suppose the number replaced by n ′ is n ′′ , then we move n ′′ up by one row to replacethe largest number smaller than n ′′ , and so on, until we replaced a number in the firstrow. (iv) Denote the number been replaced from the first row by w ( n ) , and denote the resultingtableau by P ′ . (v) Repeat the above process for n − with the tableaux pair ( P ′ , Q ′ ) , and so on, until wefind all w ( n ) , w ( n − , ..., w (1) . Then w ( P, Q ) := (cid:18) · · · nw (1) w (2) · · · w ( n ) (cid:19) . One often use the word notation w ( P, Q ) = w (1) ...w ( n ) .Proof. See e.g. [Knu98, 5.1.4]. (cid:3)
Besides the above two ingredients, we also need the Young tableaux labelling of compo-nents of the Springer fibre B u given in [Spa76] and [Ste76]. We follow Steinberg’s descriptionin [Ste88]. Proposition 3.6 (Tableaux labelling of components) . There is a bijection T ( λ ( u )) ←→ { components of B u } . More explicitly, for a given tableau P ∈ T ( λ ( u )) , the corresponding component is charac-terised as the closure of an open subset C ( P ) , where C ( P ) consists of the flags F : V ⊂ V ⊂ ... ⊂ V n = V constructed via the following steps: (I) Let N := u − I ∈ gl n , the nilpotent element associated with u ; (II) if n is in the position ( i, j ) = ( i, r i ) = ( c j , j ) of P , then V n − is any hyperplanesatisfying that ( N V n + Ker N j − ⊆ V n − N V n + Ker N j * V n − ;(III) once such a V n − is chosen, we repeat the above process to construct a V n − (byreplacing V n , N... by V n − , N | V n − ), and so on.Proof. See [Ste88, Section 2]. (cid:3)
Moreover, they proved the following property of generic relative position for the compo-nents of B u : Proposition 3.7 (Generic relative position) . Let P and Q be two standard λ ( u ) -tableaux.Then there is an open dense subsvariety X ⊆ C ( P ) × C ( Q ) such that any closed point ( F , F ) ∈ X has the relative position w ( P, Q ) .Proof. See [Ste88, Section 3] or [Spa82, II.9]. (cid:3) . Distribution of components
Our main theorem is:
Theorem 4.1.
Let U be the variety of unipotent elements of G . Then we have: (a) There is a canonical map β : U F −→ W ( T ) such that, for u ∈ U F , the Springer fibre B u intersects exactly one component ofthe Deligne–Lusztig variety X β ( u ) , and this component is an open dense subset of anirreducible component of B u . (b) Conversely, each component of X w , where w ∈ W ( T ) = S n is an involution of theshape w = [12 ... ][ ... ] ... [ ...n ] (the block [ ... ] means the reversion along the word), is adense open subset of a component of some Springer fibre.Proof. Proof of (a).
We first prove (a) for the Weyr form W ( u ) of u , in a constructible manner.Let us construct a special λ ( u )-tableau T which will be critical for us: This is done byfilling { , ..., n } into λ ( u ) in the way that, first fill the 1st column of λ ( u ) from up to down,and then the 2nd column of λ ( u ) from up to down, and so on. So we have(1) the i -th column of T = (cid:16)P j
00 0 ( I ) 0 ...
00 0 0 ( I ) ... ... ... ... ... ... ... ... I )0 0 ... , where the I ’s are the identity matrices of sizes c , c , ..., c r and the 0’s are the zero matricesof possibly different suitable sizes. Note that N V c + ... + c i ⊆ V c + ... + c i − for any flag F : V ⊆ ... ⊆ V n in C . Thus by Proposition 3.6, to show that C ⊆ C ( T ), it issufficient to show that every F ∈ C satisfies(vii) If s is in the interval ( c + ... + c i − , c + ... + c i ), then ( Ker N i − | V s +1 ⊆ V s Ker N i | V s +1 * V s ;(viii) if s = c + ... + c i (with i < r ), then ( Ker N i | V s +1 ⊆ V s Ker N i +1 | V s +1 * V s . Actually this is clear, because a direct blocked-matrix computation gives thatKer N i = V c + ... + c i for ∀ i ∈ { , ..., r } . So C ⊆ C ( T ).Next we show that B W ( u ) does not intersect any other component of X w ( T , T ) , namely, C = B W ( u ) ,w ( T , T ) . By the decomposition (3), it is sufficient to show that, if a flag F : V ⊆ ... ⊆ V n = V is stabilised by u (or equivalently, N V i +1 ⊆ V i , ∀ i ) and satisfies the condition (ii), then itautomatically satisfies the condition (v), i.e. V c + ... + c i = h e , ..., e c + ... + c i i for all i . We firstprove this for V c . Note that, for any z ∈ { , ..., c − } , the condition (ii) implies that thereis an internal direct sum decomposition(4) V c = V z ⊕ F V c − z , which gives an internal direct sum decomposition of V z +1 V z +1 = V z ⊕ F h v ′ i , where v ′ is some non-zero vector in V c − z . So by N V z +1 ⊆ V z we get(5) N V z +1 = N V z + N F h v ′ i ⊆ V z . ince N is F -stable, (5) implies that F N v ′ = N F v ′ ∈ V z ;however, by the internal decomposition (4), this happens if and only if N v ′ = 0. Thus from(5) we see that N V z +1 = N V z for any z ∈ { , ..., c − } . Therefore N V c = N V c − = ... = N V = 0 , which gives that V c = h e , e , ..., e c i . Now, suppose V c + ... + c i = h e , ..., e c + ... + c i i for some i ,then by applying the above argument to the quotient space V c + ... + c i +1 /V c + ... + c i we also get V c + ... + c i +1 = h e , ..., e c + ... + c i +1 i . So by induction the condition (v) holds, and we concludethat B W ( u ) intersects X w ( T , T ) at exactly the component C , or more precisely,(6) C = B W ( u ) ,w ( T , T ) = C ( T ) ∩ X w ( T , T ) . It remains to discuss the openness of C in the component C ( T ). First note that, sinceour N is F -stable, the step (II) in Proposition 3.6 implies that F C ( T ) = C ( T ) . So there is a graph embedding(Id , F ) : C ( T ) −→ C ( T ) × C ( T ) ⊆ G/B × G/B.
It is well-known that (see e.g. [Car93, 7.7]) the subsets D v ⊆ G/B × G/B of pairs at variousrelative positions v give a finite stratification of G/B × G/B into locally closed subvarieties,so their intersections with C ( T ) × C ( T ) also give such a stratification of C ( T ) × C ( T ); letus denote the unique dense open strata of C ( T ) × C ( T ) by X . Then Proposition 3.7 impliesthat every pair in X is in the relative position w ( T , T ). In particular we see that(Id , F ) − ( X ) = C ( T ) ∩ X w ( T , T ) , and that this is an open subvariety of C ( T ) (and hence of C ( T )). So, as X w ( T , T ) is puredimensional (see Proposition 2.3), by (6) we conclude that C is a dense open subvariety of C ( T ) as desired.The above proves (a) for W ( u ). For a general unipotent u ∈ G F = GL n ( F q ), by Proposi-tion 3.4 we know that u g = W ( u ) , for some g ∈ G . Meanwhile, as both u and W ( u ) are in G F , by the fact that two matricessimilar over a field extension are similar over the original field, we can take g ∈ G F . Thus gC is still a component of X w ( T , T ) and is also an open dense subset of the component gC ( T ) ⊆ B g W ( u ) = B u . The uniqueness of the component is also clear by taking a conjugation. This completes theproof of (a).
Proof of (b).
For such an involution w ∈ W ( T ), by Proposition 3.5 we see that w ( T , T ) = w for sometableau T of the form specified in (1). Let u be any unipotent element making T a λ ( u )-tableau. Then, as we did in the proof of (a), there is a component C of X w lying as an opendense subvariety in the component C ( T ) of B W ( u ) . ow, from the discussion of X w ( T , T ) given in (a) we know that the translation action of G F on X w is transitive on the components; in particular, each component of X w is of theform gC for some g ∈ G F , which is then an open dense subset of some component of B g W ( u ) .This completes the proof. (cid:3) Recall the dimension formula:
Corollary 4.2.
Let u be a unipotent element, then dim Z G ( u ) = X i c i , where c i is the number of boxes in the i -th column of the Young diagram of u . This can be proved by a matrix manipulation together with a combinatorial consideration(see e.g. [Hum95, 1.2 and 1.3]); here we derive a geometric proof from the argument ofTheorem 4.1.
Proof.
By Proposition 3.4 we can assume that u = W ( u ). Then from the proof of Theo-rem 4.1 we see that dim B u = dim X w ( T , T ) . By Proposition 2.3 and Proposition 2.5 this equality can be written as n ( n − −
12 dim C ( u ) = l ( w ( T , T )) , where C ( u ) denotes the conjugacy class of u . Note that, by the component decompositionin the argument of Theorem 4.1 (see (3)), we have l ( w ( T , T )) = X i l ( R i ) = X i c i ( c i − . So dim Z G ( u ) = dim G − dim C ( u ) = n + X i c i ( c i −
1) = X i c i as desired. (cid:3) Examples and remarks
In this section we give some examples and remarks related to our main theorem, and givea short discussion on the representations associated with B u,w = B u ∩ X w for rectangular u .1.One may wonder that, in Theorem 4.1, does (a) applies to an arbitrary component of B u , or does (b) applies to an arbitrary involution of W ( T )? As illustrated below, withoutweakening the assertions usually the answer is no: Example 5.1.
Let G = GL and let u = W ( u ) = ∈ GL ( F q ) . y Proposition 2.5 and Proposition 3.6, the variety B u has two irreducible components, bothof dimension 2, indexed by the Young tableaux P := 1 32 4 and Q := 1 23 4 , respectively. The Robinson–Schensted correspondence gives two involutions, w ( P, P ) = 2143 = (1 , , w ( Q, Q ) = 3412 = (2 , , , , . (The second equalities are for writing the elements as reduced products of simple reflections.)First consider the component C ( P ). Explicitly, by computing the flag condition (*) for theDeligne–Lusztig variety X , and by computing the step (II) of Proposition 3.6 for C ( P ),one can see that X ∼ = Gr(2 , F × (cid:0) P \ P ( F q ) (cid:1) and C ( P ) ∼ = ( P ) , which suggests that C ( P ) contains a component of X as a dense open subset, and fromthe argument of Theorem 4.1 we know that this is exactly the case, and this component isgiven as B u, = B u ∩ X .Now consider the other component C ( Q ). As mentioned in the argument of Theorem 4.1,Proposition 3.7 implies that the variety of pairs of flags at relative position w = 3412 cutout an open dense subset X of C ( Q ) × C ( Q ), and so the preimage C ( Q ) ∩ X of X alongthe Frobenius graph embedding(Id , F ) : C ( Q ) −→ C ( Q ) × C ( Q )is an open subset of C ( Q ); indeed, this open subset is non-empty: By fixing a standard basis { e , e , e , e } of V = F q (over F q ), one easily checks that it contains the flag { } ⊆ { e + xe } ⊆ { e + xe , e + xe } ⊆ { e , e , e + xe } ⊆ V for any x ∈ F q \ F q . This means that some component of X cuts out an open dense subsetof C ( Q ). However, sincedim X = l ((2 , , , , > B u , and since the Deligne–Lusztig varieties partition the flag variety, the component C ( Q ) cannotcontain any component of a Deligne–Lusztig variety as an open dense subset. On the otherhand, by Proposition 2.5 and Corollary 4.2 we see that the possible dimensions of Springerfibres for GL are 0 , , , ,
6, so none of the components of X can be an open subset ofa Springer fibre.2.It is a natural desire that, the map β ( − ) is uniquely characterised by the property givenin Theorem 4.1. To make this true one shall add further conditions: xample 5.2. Let G = GL and let u = W ( u ) = ∈ GL ( F q ) . Then the components of B u are labelled by the two tableaux of hook shape P := 1 32 and Q := 1 23 , corresponding to the simple reflections (1 ,
2) and (2 , B u , X (1 , , and X (2 , , are all of dimension 1.From the argument of Theorem 4.1 we know that C ( P ) contains a component of X (1 , asa dense open subset and B u does not intersect other components of X (1 , . Let us consider X (2 , . Given a flag V ⊆ V ⊆ V ⊆ V ⊆ V , by direct computations with the flag condition (*) and the condition that N = u − I takes V i into V i − , one sees that B u intersects X (2 , only at the component (of X (2 , ) consisting of the flags { } ⊆ { e } ⊆ { e , e + λe } ⊆ { e , e , e } = V, where λ runs over F q \ F q . Meanwhile, it follows from the step (II) of Proposition 3.6 thatthis component is contained in C ( Q ). So, at this u , there are two choices of the value of β fulfilling the requirement in Theorem 4.1.Thus we hope to state here a question: Question 5.3.
Is there a geometric property (in addition to the one asserted in Theorem 4.1,but without referencing to the explicit construction (2) given in its argument) making themap β unique?3.Let w be the longest element of W ( T ), then X w contributes a generic (i.e. open dense)part of the flag variety. Quite opposite to the component containment relation in Theo-rem 4.1, Springer fibres missed this “largest” Deligne–Lusztig variety at all: Example 5.4.
In this example we show that B u,w = B u ∩ X w is always empty unless u = 1.Suppose that B u,w is non-empty; let F ∈ B u,w be a point in the component C ( P ) for some λ ( u )-tableau P . Consider the Frobenius graph embedding(Id , F ) : C ( P ) −→ C ( P ) × C ( P ) . (Here F preserves C ( P ) because F preserves C ( P ) and F is a homeomorphism.) By Propo-sition 3.7 and by considering Bruhat order we get“the relative position of ( F , F F )” = w ≤ w ( P, P ) , which in turn implies that w ( P, P ) = w . So, via the Robinson–Schensted correspondencewe see that λ ( u ) must be a single column diagram, that is, u = 1, in which case B u is thewhole flag variety. .Recall that (see Notation 3.1) we have let J ( u ) be the Jordan normal form of u , d thenumber of Jordan blocks in J ( u ), r i the sizes of each Jordan block, and λ ( u ) the associatedYoung diagram. Definition 5.5.
A unipotent element u ∈ G is called rectangular, if the Young diagram λ ( u ) is rectangular.Note that if u is rectangular, then d (= c i , ∀ i ) is a divisor of n and n/d = r i for all i ; whenthis is the case we denote r i by r .As mentioned in the introduction, our original focus on the relations between Deligne–Lusztig varieties and Springer fibres comes from the smooth representation theory of theprofinite group GL d ( F q [[ π ]]). Indeed, if u ∈ U F is rectangular, then the finite quotientgroup GL d ( F q [[ π ]] /π r ) acts on B u,w , because GL d ( F q [[ π ]] /π r ) is naturally isomorphic to the G -centraliser of W ( u ) = I d I d ... I d I d ...
00 0 I d ... ... ... ... ... ... ... ... I d I d ... I d , where I d denotes the d × d identity matrix, by the ring injection from M d ( F q [[ π ]] /π r ) to M n ( F q ):(7) A + A π + ... + A r − π r − A A ... A r − A r − A A ... A r − ... ... ... ... ... ... A A ... A , where A i ∈ M d ( F q ). (See [Che20a, 5.3] and [Che20b, 4.6]; note that the notation used thereis different up to a blocked transpose.) Thus we get a virtual representation R u,w := X i ( − i H ic ( B u,w , Q ℓ )of GL d ( F q [[ π ]] /π r ). Here H ic ( − , Q ℓ ) denotes the i -th compactly supported ℓ -adic cohomologywith ℓ a prime not equal to char( F q ). Note that if n = d , then this construction gives theunipotent representations of GL n ( F q ) in the sense of [DL76, 7.8]. Definition 5.6.
A representation of GL d ( F q [[ π ]] /π r ), where r ≥
2, is called primitive, if itdoes not factor through GL d ( F q [[ π ]] /π r − ).For example, when u ∈ U F is rectangular with d, r ≥
2, from [Che20b, 4.8] we known that R W ( u ) ,w is a primitive representation of GL d ( F q [[ π ]] /π r ) if w = (1 , ..., z ) is a cycle with z < d . Question 5.7.
For a given rectangular u ∈ U F (resp. Weyl element w ∈ W ( T )), is there areasonable characterisation of the Weyl element w (resp. rectangular u ∈ U F ) making R u,w primitive? e note that there is a uniform description of the representations R u,w for various rectan-gular u ∈ U F , using a complex on G : First consider the character-sheaf type diagram (see[Lus85]) X w Z w G, b a where Z w := { ( g, xB ) ∈ G × L − ( BwB ) /B | g x ∈ B } , and a, b are the natural projections; this diagram is G F -equivariant, where G F acts on Z w by h · ( g, x ) = ( hgh − , hx ), on G by left conjugation, and on X w by left multiplication. Thenthe character-sheaf type complex K := Ra ! b ∗ Q ℓ ∈ D bc ( G, Q ℓ )encodes all R u,w : Proposition 5.8. If u ∈ U F is rectangular, then R u,w = P i ( − i H i ( K ) u .Proof. Note that B u,w ∼ = a − ( u ), so the assertion formally follows from the proper basechange along { u } ֒ → G . (cid:3) We end with a (non-)smoothness property of the above diagram.
Proposition 5.9.
The variety Z w and the morphism b are smooth. However, the morphism a is smooth only for n = 1 .Proof. We follow the idea of [Lus85, 2.5.2] to use a faithful flat descent. Let f Z w be the basechange of Z w ⊆ G × L − ( BwB ) /B along the morphism G × L − ( BwB ) −→ G × L − ( BwB ) /B. This is faithfully flat since B is solvable (hence the quotient L − ( BwB ) → L − ( BwB ) /B admits local sections). Then by [GD67, 17.7.7] it suffices to prove that the variety f Z w = { ( g, x ) ∈ G × L − ( BwB ) | g x ∈ B } ⊆ G × G and the morphism e b : f Z w → X w (extending b )are smooth. Applying the variable change y = g x we get f Z w ∼ = { ( y, x ) ∈ G × L − ( BwB ) | y ∈ B } = B × L − ( BwB ) , and e b reads as ( y, x ) xB . As the Lang morphism L is ´etale, the assertion on Z w and b follows.For the assertion on a , again by the faithful flat descent it suffices to show that themorphism f Z w → G given by e a : ( y, x ) ∈ B × L − ( BwB ) xyx − is not smooth. Actually this morphism is not flat: Note that the fibre of e a at any centralelement c ∈ Z ( G ) is { c } × L − ( BwB ) ⊆ B × L − ( BwB ). So, if e a is flat then we have (seee.g. [Liu06, 4.3.12]) dim L − ( BwB ) = dim B + dim L − ( BwB ) − dim G, which is impossible unless n = 1. (cid:3) n the above argument, note that for any b ∈ B , the elements ( y, x ) and ( b − yb, xb ) havethe same image under e a . So, the fibre of e a at a closed point is either empty or has dimensionat least dim B , which implies that (via [Liu06, 4.3.12])dim B ≤ e a − ( xyx − ) = dim B + dim L − ( BwB ) − dim G if e a is flat at ( y, x ). Thus actually e a cannot be flat at any point unless w is the longestelement w . References [Car93] Roger W. Carter.
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Department of Mathematics, Shantou University, Shantou, China
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