Algebraically Independent Generators for the Algebra of Invariant Differential Operators on SL n (R)/ SO n (R)
aa r X i v : . [ m a t h . R T ] O c t Algebraically Independent Generators for the Algebra ofInvariant Differential Operators on SL n ( R ) / SO n ( R ) Dominik Brennecken, Lorenzo Ciardo and Joachim Hilgert
Abstract.
We provide an explicit set of algebraically independent generators forthe algebra of invariant differential operators on the Riemannian symmetric spaceassociated with SL n ( R ). Mathematics Subject Classification 2020:
Key Words and Phrases:
Invariant differerential operators, Riemannian symmetricspaces, Maass-Selberg operators, symmetric cones.
1. Introduction
Let Pos n ( R ) = GL n ( R ) / O n ( R ) be the space of positive definite real n × n -matricesand SPos n ( R ) = SL n ( R ) / SO n ( R ) its subset of elements of determinant 1. Bothspaces are Riemannian symmetric spaces whose algebras D (Pos n ( R )), respectively D (SPos n ( R )), of invariant differential operators are commutative. The Harish-Chandra isomorphism together with Chevalley’s Theorem shows that these alge-bras are isomorphic to the polynomial algebras in n , respectively n −
1, variables.For D (Pos n ( R )) one finds various explicit sets of algebraically independent gener-ators in the literature. One example of such a set is given by the Maass-Selbergoperators δ , . . . , δ n described in detail in [9]. It is much harder to track down anexplicit set of algebraically independent generators for D (SPos n ( R )) in the litera-ture. In [6, §
2] one finds a set of algebraically independent generators for the center Z of the universal enveloping algebra of sl n ( C ), which is known to be isomorphicto the algebra of invariant differential operators on SL n ( R ) / SO n ( R ) (see e.g. [7,Prop. II.5.32 & Exer. II.D.3]). In [11, §
7] the author follows a similar line giving thegenerators (without proof) as coefficients in a characteristic polynomial.In this paper we show how to derive algebraically independent generators for D (SPos n ( R )) from the Maass-Selberg operators as the SPos n ( R )-parts (constructedin Lemma 3.2) of the Maass-Selberg operators δ , . . . , δ n (Theorem 4.3). The argu-ments given can be applied in more generality as we will explain in Section 5. Thekey input of the paper comes from the master theses [2, 3] written by the first twoauthors under the direction of the third author. Brennecken, Ciardo and Hilgert
2. Preliminaries on Invariant Differential Operators
For a real reductive Lie group G in the Harish-Chandra class [5] and a maximalcompact subgroup K ≤ G we denote by D K ( G ) = { D ∈ D ( G ) | ∀ k ∈ K : Ad( k ) D = D } the K -invariant elements of D ( G ), i.e., the differential operators that are G -invariantfrom the left and K -invariant from the right. We also declare D ( G ) k = h D ˜ X | D ∈ D ( G ) , X ∈ k i C − vector space to be the left ideal generated by k in D ( G ). Here ˜ X is the left-invariant vector fieldon G associated with X . Proposition 2.1 ([7, Thm. II.4.6]) . Let π : G → G/K be the canonical projection.Then the map µ : D K ( G ) → D ( G/K ) , ( µ ( D ) f )( gK ) = D ( f ◦ π )( g ) , f ∈ C ∞ ( G/K ) , g ∈ G is an algebra epimorphism with kernel D K ( G ) ∩ D ( G ) k . Moreover, we have anisomorphism of algebras µ : D K ( G ) / ( D K ( G ) ∩ D ( G ) k ) → D ( G/K ) . There exists a unique linear isomorphism, called the symmetrization of D ( G ), λ G = λ : S ( g C ) → D ( G )satisfying λ ( X m ) = ˜ X m for each X ∈ g , where S ( g C ) is the symmetric algebra of g C .If X , . . . , X n is a basis of g , then under the identification C [ X , . . . , X n ] ∼ = S ( g C ),we obtain for each P ∈ S ( g C ) λ ( P ) f ( g ) = P (cid:18) ∂∂t , . . . , ∂∂t n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( g exp( t X + . . . + t n X n )) , f ∈ C ∞ ( G ) , g ∈ G, (1)where the suffix | t =0 means the evaluation in t = ( t , . . . , t n ) = 0 after differentiation([7, Thm. II.4.3]). Furthermore, for any Y , . . . , Y r ∈ g we have λ ( Y · · · Y r ) = 1 r ! X σ ∈S r ˜ Y σ (1) · · · ˜ Y σ ( r ) , (2)where S r is the symmetric group of permutations on r elements.Let g = k ⊕ p be the Cartan decomposition and I ( p C ) the space of complex K -invariant polynomials on p C . Proposition 2.2 ([7, Thm. II.4.9]) . The map µ ◦ λ G : I ( p C ) → D ( G/K ) , P D P = µ ( λ G ( P )) is a linear isomorphism. Moreover, for each P , P ∈ I ( p C ) there exists a Q ∈ I ( p C ) with degree smaller than the sum of the degrees of P , P and D P P = D P D P + D Q . rennecken, Ciardo and Hilgert P , . . . , P n are generators of I ( p C ), then D P , . . . , D P n are generators of D ( G/K ). Remark 2.3. If D ∈ D ( G/K ) and X , . . . , X r is a basis of p , then there existsa polynomial P ∈ S ( p C ), such that Df ( gH ) = P (cid:18) ∂∂t , . . . , ∂∂t r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t =0 f ( g exp( t X + . . . + t r X r ) K ) , f ∈ C ∞ ( G/K ) , which is uniquely determined, since ( t , . . . , t r ) g exp( t X + . . . + t r X r ) K is alocal chart around gK . Thus, Proposition 2.2 implies that P is automatically K -invariant, i.e., P ∈ I ( p C ).
3. Maass-Selberg Operators and their SPos n ( R ) -Radial Parts Let Sym n ( R ) be the set of symmetric n × n -matrices. Then the k -th Maass-Selbergoperator δ k ∈ D (Pos n ( R )) is given by δ k f ( g O n ( R )) = tr (cid:18) ∂∂X (cid:19) k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f ( g exp( X )O n ( R )) , f ∈ C ∞ (GL n ( R ) / O n ( R )) , where ∂∂X = ∂∂x . . . ∂∂x n ... . . . ... ∂∂x n . . . ∂∂x nn and X = x . . . x n ... . . . ... x n . . . x nn ∈ Sym n ( R ) . To construct the SPos n ( R )-parts of the δ k we consider the maps i : SL n ( R ) / SO n ( R ) → GL n ( R ) / O n ( R ) , g SO n ( R ) g O n ( R )and p : GL + n ( R ) / SO n ( R ) → SL n ( R ) / SO n ( R ) , g O n ( R ) det( g ) − n g SO n ( R ) , where GL + n ( R ) = { g ∈ GL n ( R ) | det( g ) > } . Note thatGL + n ( R ) / SO n ( R ) → GL n ( R ) / O n ( R ) , g SO n ( R ) g O n ( R )is a diffeomorphism. We use it to identify the two spaces. For g ∈ SL n ( R ) let τ g denote the translation by g on GL n ( R ) / O n ( R ) and SL n ( R ) / SO n ( R ). Proposition 3.1.
The maps i and p are SL n ( R ) -equivariant, i.e. ∀ g ∈ SL n ( R ) : p ◦ τ g = τ g ◦ p and i ◦ τ g = τ g ◦ i. Proof.
Let h O n ( R ) ∈ GL n ( R ) / O n ( R ) and g ∈ SL n ( R ). We can assume thatdet( h ) >
0, so p ( g.h O n ( R )) = det( gh ) − n gh SO n ( R ) = g. det( h ) − n h SO n ( R ) = g.p ( h O n ( R )) . Brennecken, Ciardo and Hilgert
For h SO n ( R ) ∈ SL n ( R ) / SO n ( R ) we have i ( g.h SO n ( R )) = gh O n ( R ) = g.h O n ( R ) = g.i ( h SO n ( R )) . Lemma 3.2.
The map P : D (GL n ( R ) / O n ( R )) → D (SL n ( R ) / SO n ( R )) defined by P ( D ) f = D ( f ◦ p ) ◦ i for f ∈ C ∞ (SL n ( R ) / SO n ( R )) is a morphism of algebras. Furthermore, it satisfies ( P ( D ) f ) ◦ p = D ( f ◦ p ) (3) for D ∈ D (GL n ( R ) / O n ( R )) and f ∈ C ∞ (SL n ( R ) / SO n ( R )) . We call P ( D ) the SPos n ( R ) -radial part of D . Proof.
First we check that the image of P consists of invariant differential oper-ators on SL n ( R ) / SO n ( R ). If f ∈ C ∞ (SL n ( R ) / SO n ( R )), then f ◦ p is smooth since p is. As D is a differential operator and i is smooth, we have P ( D ) f = D ( f ◦ p ) ◦ i ∈ C ∞ (SL n ( R ) / SO n ( R )). The linearity of P ( D ) is an easy consequence of the linearityof D . Moreover, D satisfies supp( D ( f ◦ p )) ⊆ supp( f ◦ p ). Precomposing by i wefind supp( P ( D ) f ) = supp( D ( f ◦ p ) ◦ i ) ⊆ supp( f ◦ p ◦ i ) = supp( f ) , so that Peetre’s Theorem shows that P ( D ) is a differential operator on SL n ( R ) / SO n ( R ).From the invariance of D ∈ D (GL n ( R ) / O n ( R )) and Proposition 3.1 we obtain P ( D )( f ◦ τ g ) = D ( f ◦ τ g ◦ p ) ◦ i = D ( f ◦ p ) ◦ i ◦ τ g = P ( D ) f ◦ τ g for g ∈ SL n ( R ), so P ( D ) ∈ D (SL n ( R ) / SO n ( R )).The linearity of P is clear. Next we prove equation (3). Let g O n ( R ) ∈ GL n ( R ) / O n ( R ) be arbitrary and assume that det( g ) >
0. Define h = det( g ) − n n .Then by the invariance of D ∈ D (GL n ( R )) / O n ( R )) and Proposition 3.1 we obtain( P ( D ) f ◦ p )( g O n ( R )) = ( D ( f ◦ p ) ◦ i )(det( g ) − n g SO n ( R ))= D ( f ◦ p )(det( g ) − n g O n ( R ))= ( D ( f ◦ p ) ◦ τ h )( g O n ( R ))= D ( f ◦ p ◦ τ h )( g O n ( R )) . Since τ h only scales the determinant we have p ◦ τ h = p , this implies (3).Let D , D ∈ D (GL n ( R ) / SL n ( R )). Together with equation (3), we conclude P ( D D ) f = ( D D ( f ◦ p )) ◦ i = D ( P ( D ) f ◦ p ) ◦ i = P ( D ) P ( D ) f. As P (id D (GL n ( R ) / O n ( R ) )) = id D (SL n ( R ) / SO n ( R )) is obvious, this concludes the proof. rennecken, Ciardo and Hilgert D ∈ D (GL n ( R ) / O n ( R )) thereexists a polynomial Q D on Sym n ( R ), such that Df ( g O n ( R )) = Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f ( g exp( X )O n ( R )) , where X ∈ Sym n ( R ) and f ∈ C ∞ (GL n ( R ) / O n ( R )). This leads to another represen-tation of the morphism P . Proposition 3.3.
For each f ∈ C ∞ (SL n ( R ) / SO n ( R )) P ( D ) f ( g SO n ( R )) = Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f (cid:18) g exp (cid:18) X − n tr( X ) n (cid:19) SO n ( R ) (cid:19) . Proof. P ( D ) f ( g SO n ( R )) = D ( f ◦ p )( g O n ( R )))= Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f ( p ( g exp( X )O n ( R )))= Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f (det(exp( X )) − / n g exp( X )SO n ( R ))= Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f (cid:18) g exp (cid:18) − n tr( X ) n (cid:19) exp( X )O n ( R ) (cid:19) = Q D (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f (cid:18) g exp (cid:18) X − n tr( X ) n (cid:19) · SO n ( R ) (cid:19) . In fact, the next lemma shows that P is surjective. Thus, the images P ( δ ) , . . . , P ( δ n ) of the Maas-Selberg operators are generators of D (SPos n ( R )) andwe will show that P ( δ ) = 0, so that it remains to show that P ( δ ) , . . . , P ( δ n ) arealgebraically independent. Lemma 3.4.
The morphism P is surjective and P ( δ ) = 0 . Proof.
Let D ∈ D (Pos n ( R )) be given by a unique polynomial Q on Sym n ( R ),i.e. Df ( g O n ( R )) = Q (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f ( g exp( X )O n ( R ))holds for every f ∈ C ∞ (GL n ( R ) / O n ( R )). Denote by h X, Y i = tr( XY ) the innerproduct on Sym n ( R ). Then it is easy to verify that for A ∈ Sym n ( R ) we have ∂∂x ii e h X,A i = a ii e h X,A i , ∂∂x ij e h X,A i = a ij e h X,A i . (4)Let Q be the unique polynomial on SSym n ( R ) so that for each function f ∈ C ∞ (SL n ( R ) / SO n ( R )) we have P ( D ) f ( g SO n ( R )) = Q (cid:18) ∂∂Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Y =0 f ( g exp( Y )SO n ( R )) . Brennecken, Ciardo and Hilgert
We associate to a matrix A ∈ SSym n ( R ) the smooth function f A : GL n ( R ) / O n ( R ) → R , exp( Y )SO n ( R ) e h Y,A i with Y ∈ SSym n ( R ) . We can observe with Proposition 3.3 and equation (4) that Q ( A ) = Q (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 e h X,A i = Q (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 e tr (( X − tr( X ) n n ) A )= Q (cid:18) ∂∂X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f A (cid:18) exp (cid:18) X − tr( X ) n n (cid:19) SO n ( R ) (cid:19) = P ( D ) f A ( n SO n ( R ))= Q (cid:18) ∂∂Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Y =0 f A (exp( Y )SO n ( R ))= Q (cid:18) ∂∂Y (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Y =0 e h Y,A i = Q ( A ) . Thus Q is the restriction of Q to SSym n ( R ), so P is surjective. Moreover, for D = δ we have Q = tr and therefore Q = 0, hence P ( δ ) = 0.To summarize, we have shown the following theorem. Theorem 3.5.
Let δ , . . . , δ n be the Maass-Selberg operators. Then P ( δ ) = 0 and P ( δ k ) f ( g SO n ( R )) = tr (cid:18) ∂∂X (cid:19) k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X =0 f (cid:18) g exp (cid:18) X − n tr( X ) n (cid:19) · SO n ( R ) (cid:19) , where ∂∂X = ∂∂x . . . ∂∂x n ... . . . ... ∂∂x n . . . ∂∂x nn and X = x . . . x n ... . . . ... x n . . . x nn ∈ Sym n ( R ) are generators for the algebra D (SL n ( R ) / SO n ( R )) . Using the proof of Lemma 3.4 we can express the P ( δ k ) in terms of a localchart for SPos n ( R ) = SL n ( R ) / SO n ( R ). Corollary 3.6.
For f ∈ C ∞ (SL n ( R ) / SO n ( R )) and Y ∈ SSym n ( R ) we have P ( δ k ) f ( g SO n ( R )) = tr (cid:18) ∂∂Y (cid:19) k !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Y =0 f ( g exp( Y ) · SO n ( R )) . (5)It is possible to prove the surjectivity of P by giving an explicit algebrasplitting. In fact, for g SO n ( R ) ∈ GL + n ( R ) / SO n ( R ) one can define the map ϕ g : GL + n ( R ) / SO n ( R ) → GL + n ( R ) / SO n ( R ) , h SO n ( R ) → det( g ) n h SO n ( R ) , rennecken, Ciardo and Hilgert g O n ( R ). Using ϕ g and the identification GL n ( R ) / O n ( R ) =GL + n ( R ) / SO n ( R ) one obtains an algebra morphism I : D (SPos n ( R )) → D (GL n ( R ) / O n ( R ))via I ( D ) f ( g O n ( R )) = ( D ( f ◦ ϕ g ◦ i ) ◦ p )( g O n ( R ))for f ∈ C ∞ (GL n ( R ) / O n ( R )) and D ∈ D (SL n ( R ) / SO n ( R )). This morphism satisfies I ( D ) f ◦ i = D ( f ◦ i ) , P ◦ I = id D (SL n ( R ) / SO n ( R )) . For more details we refer to [2, Lemma 4.5, Theorem 4.6].
4. Algebraic Independence
In this section we prove the algebraic independence of P ( δ ) , . . . , P ( δ n ). We startwith a general lemma on polynomial algebras. Lemma 4.1.
Let k be a field and A an (associative) commutative unital algebraover k . Assume that A contains algebraically independent generators x , . . . , x n ∈ A . Then(i) If y , . . . , y m ∈ A generate A , then m ≥ n .(ii) If y , . . . , y n ∈ A generate A , then y , . . . , y n are algebraically independent.(iii) The number n is independent of the choice of algebraically independent gener-ators. Proof.
Since x , . . . , x n ∈ A are algebraically independent generators of A , wehave an algebra isomorphism k [ X , . . . , X n ] → A, X i x i . So we may assume that A = k [ X , . . . , X n ] is the ring of polynomials in n -indeterminants and x i = X i .(i) We define a morphism of algebras by k [ Y , . . . , Y m ] → k [ X , . . . , X n ] , Y i y i . The morphism ϕ is surjective, because y , . . . , y m are generators of A = k [ X , . . . , X n ]. If dim denotes the Krull-dimension (cf. [1, Def. 2.5.3]), then thesurjectivity of ϕ implies dim( k [ Y , . . . , Y m ]) ≥ dim( k [ X , . . . , X n ]) as preimagesof prime ideals are prime (cf. [1, Prop. 2.5.5]). Now [1, Corollary 2.25.2] impliesdim( k [ Y , . . . , Y m ]) = m and dim( k [ X , . . . , X n ]) = n , and hence the claim.(ii) If y , . . . , y n are generators of A = k [ X , . . . , X n ], then it suffices to show thatthe map ϕ from part (i) is injective, because the Y , . . . , Y n are algebraicallyindependent.Assume that 0 = ker( ϕ ). Then we can find a polynomial 0 = f ∈ ker( ϕ ).Decompose f = f · · · f r into irreducible polynomials f i . Then0 = ϕ ( f ) = ϕ ( f ) · · · ϕ ( f r ) Brennecken, Ciardo and Hilgert leads to an irreducible polynomial f i ∈ ker( ϕ ), since k [ X , . . . , X n ] has no zerodivisors. Let I ⊆ ker( ϕ ) be the ideal generated by f i . Moreover, I is a primeideal which does not contain any proper prime ideal except { } . This meansthat the height ht( I ) of I is 1 (cf. [1, Def. 2.5.4 & Prop. 2.25.3]), so that [1,Lemma 2.25.7] implies dim( k [ Y , . . . , Y n ] /I ) = n −
1. The morphism ϕ factorsthrough I to a surjective morphism ˜ ϕ : k [ Y , . . . , Y n ] /I → k [ X , . . . , X n ].Again, we conclude dim( k [ Y , . . . , Y n ] /I ) ≥ dim( k [ X , . . . , X n ]) and obtain acontradiction proving the claim.(iii) This follows applying parts (i) and (ii) to two different algebraically indepen-dent generators x , . . . , x n and x ′ , . . . , x ′ m of A . Proposition 4.2.
Let δ , . . . , δ n be the Maass-Selberg operators. Then the oper-ators P ( δ ) , . . . , P ( δ ) are algebraically independent. Proof.
By the Harish-Chandra Isomorphism and Chevalley’s Theorem about in-variants of finite reflection groups, we know that D (SL n ( R ) / SO n ( R )) is generatedby n − δ , . . . , δ n are generators of D (GL n ( R ) / O n ( R )) and P : D (GL n ( R ) / O n ( R )) → D (SL n ( R ) / SO n ( R )) is an epimor-phism of algebras and P ( δ ) = 0 by Theorem 3.5, the operators P ( δ ) , . . . , P ( δ n )generate D (SL n ( R ) / SO n ( R )). Finally, Lemma 4.1 implies the algebraic indepen-dence. Combining Proposition 4.2 with the results of Section 3 we obtain Theorem 4.3.
The operators P ( δ ) , . . . , P ( δ ) given by (5) form an algebraicallyindependent set of generators for D (SPos n ( R )) .
5. Related Results
The arguments given in this paper also work for different sets of algebraically inde-pendent generators for D (Pos n ( R )). Remark 5.1 ([2, Thm. 3.7&Prop. 4.13]) . Let X = ( x ij ) i,j ∈ p be a symmetricmatrix with eigenvalues λ , . . . , λ n . Assume that Y , . . . , Y n are the elementarysymmetric polynomials in n indeterminants. Then one can see that the characteristicpolynomial of X can be written asdet( t n − X ) = n Y i =1 ( t − λ i ) = t n + n X k =1 ( − k Y k ( λ , . . . , λ n ) t n − k . (6)Let F k ( X ) be the sum over the k × k principal minors of X , i.e., F k ( X ) = X ≤ i <...
Let Ω be a symmetric cone and D , . . . , D r the set of Nomura’sgenerators for D (Ω) . Then P ( D ) , . . . , P ( D r ) are algebraically independent genera-tors for D ( S Ω) . The classification of symmetric cones ([4, p. 97]) shows that this yields alge-braically independent generators for the algebras of invariant differential operatorsfor the following Riemannian symmetric spaces: SL n ( R ) / SO n ( R ), SL n ( C ) / SU n ( C ),SL n ( H ) / SU n ( H ), O ,n − ( R ) / O n − ( R ), E − /F , where H denotes the quaternions0 Brennecken, Ciardo and Hilgert and E − /F is the space of positive definite 3 × Acknowledgment:
We thank the anonymous referee for suggesting the short argumentleading to Theorem 3.5.
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