Kazhdan s orthogonality conjecture for real reductive groups Binyong Sun (Joint with Jing-Song Huang) Academy of Mathematics and Systems Science Chinese Academy of Sciences IMS, NUS March 28, 2016
Contents 1. Euler-Poincaré pairing 2. Global characters 3. Elliptic pairings 4. The elliptic Grothendieck group 5. Kazhdan s orthogonality conjecture 6. The real case 7. The proof
1. Euler-Poincaré pairing Let G be a connected reductive algebraic group over Q p. Let G V 1, V 2 be smooth representations. Assume that they have finite length.
Definition: The Euler-Poincaré pairing EP G (V 1, V 2 ) := i Z( 1) i dim Ext i G (V 1, V 2 ).
Example 1 If G has a non-compact center, then EP G (V 1, V 2 ) = 0.
Example 2 If G has a compact center, and either V 1 or V 2 is supercuspidal, then EP G (V 1, V 2 ) = dim Hom G (V 1, V 2 ).
2. Global characters The space of generalized functions: C (G) := Hom C (C 0 (G) dg, C). The space of locally integrable functions: } L 1 loc {f (G) := : f (g)ϕ(g) dg < for all ϕ /. G
Put G rs := {regular semisimple element in G}. Inclusions and restrictions: C (G) L 1 loc (G) C (G) injective injective C (G rs ) L 1 loc (G rs) C (G rs )
Let G V be a smooth representation of finite length. Its global character Θ V C (G) is defined to be ( ϕ dg the trace of V V, v G ) ϕ(g)g.v dg.
Clearly, Θ V C (G) G. Harish-Chandra s regularity theorem: Θ V L 1 loc (G) and Θ V Grs C (G rs ).
3. Elliptic pairings Definition: An element x G is elliptic if its centralizer Z G (x) in G has a compact center. Put G re := {regular elliptic element in G} and C := G re /G.
Remarks: elliptic semisimple. C is non-empty G has a compact center. We are concerned with Θ V C C (C).
The Weyl measure dc on C: f (g) dg = for all f C 0 (G re ) C 0 (G). G C G f (gcg 1 ) dg dc, Lemma (Kazhdan, 1986) Θ V C L 2 (C; dc).
Definition: The elliptic pairing (V 1, V 2 ) ell := Θ V1 (c 1 )Θ V2 (c) dc. Remarks: (V 1, V 2 ) ell = (V 2, V 1 ) ell. Θ V (c 1 ) = Θ V (c). C
Example 1 If G has a non-compact center, then (V 1, V 2 ) ell = 0.
Example 2 If either V 1 or V 2 is properly induced, then (V 1, V 2 ) ell = 0.
4. The elliptic Grothendieck group Let R(G) denote the Grothendieck group (with C-coefficients) of the category of finite length smooth representations of G. The elliptic Grothendieck group is R(G) := R(G) the span of all properly induced representations.
Lemma (Harish-Chandra) R(G) C (G), V Θ V. Lemma (Kazhdan) Assume that G has a compact center. R(G) L 2 (C; dc), V Θ V C. Remark: Assume that G has a compact center. The elliptic pairing induces an inner product on R(G).
5. Kazhdan s orthogonality conjecture Theorem (Kazhdan s orthogonality conjecture) (Schneider-Stuhler, 1997 and Bezrukavnikov, 1998) EP G (V 1, V 2 ) = (V 1, V 2 ) ell. The same holds in the case of real groups.
6. The real case Let G be a real reductive group in Harish-Chandra s class: (a) it has only finitely many connected components, and its Lie algebra g 0 is reductive; (b) it has a connected closed subgroup with finite center whose Lie algebra equals [g 0, g 0 ]; (c) for every g G, the adjoint action Ad(g) : g g is an inner automorphism of g, where g denotes the complexification of g 0.
Let K be a maximal compact subgroup of G. Let (g, K) V 1, V 2 be two modules. Assume that they have finite length. The Euler-Poincaré pairing EP g,k (V 1, V 2 ) := i Z( 1) i dim Ext i g,k (V 1, V 2 ).
Define C := G re /G and dc as in the p-adic case. Remark: C is non-empty rankg = rankk.
For every (g, K) V of finite length, the global character Θ V C (G) is defined; Θ V L 1 loc (G); Θ V Grs C (G rs ); Θ V C L 2 (C).
The elliptic pairing: (V 1, V 2 ) ell := C Θ V1 (c 1 )Θ V2 (c) dc. Remark: The elliptic pairing (V 1, V 2 ) ell only depends on V 1 K and V 2 K.
Theorem (Huang-Sun, arxiv 2015) EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell.
Corollary If rankg rankk, then EP g,k (V 1, V 2 ) = 0. Example: G = SL n (R), n 3.
Corollary The Euler-Poincaré pairing EP g,k is symmetric and non-negative. Assume that G has a compact center. It defines an inner product on R(g, K) R := the K-group, with R-coef., of f.l. (g, K)-modules. the span of properly induced representations
Remarks: If G is connected and rankg = rankk, then the Dirac pairing (V 1, V 2 ) Dir Z is defined by using the Dirac cohomologies. It only depends on V 1 K and V 2 K. It is easy to verify that (V 1, V 2 ) Dir = (V 1, V 2 ) ell. This is carried out by Huang and Renard.
7. The proof Step 1: If G has a non-compact center, then EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell = 0.
Lemma Let a be an abelian finite dimensional complex Lie algebra. If a 0, then ( 1) i dim H i (a; V ) = 0 i Z for all finite dimensional representations V of a. Proof. ( ) n n i=0( 1) i = 0, for n 1. i
Step 2: If V 2 is properly induced, then EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell = 0. Proof. Using Frobenius reciprocity, this is reduced to Step 1. Every proper Levi subgroup has a non-compact center.
then Step 3: If V 2 is in discrete series or limit of discrete series, EP g,k (V 1, V 2 ) = (V 1, V 2 ) ell. Proof. Assume that rankg = rankk, and let T K be a Cartan subgroup of G. Then V 2 is cohomologically induced from T V 0, using a Borel subalgebra b = t n. Then dim K/T ( 1) 2 EP g,k (V 1, V 2 ) = j Z( 1) j dim Hom T (H j (n; V 1 ), V 0 ).
For all t T G rs, Vogan: Θ V1 (t) = i Z ( 1)i Θ Hi (n,v )(t). i Z ( 1)i Θ i n(t) Harish-Chandra: dim K/T ( 1) 2 Θ V2 (t) = w W(T,G) Θ V0 (w.t) i Z ( 1)i Θ i n(w.t) ;
Step 4: The general case. Proof. Assume that G has a compact center. Then R(G) is generated by properly induced representations, when rankg rankk; discrete series representations, limit of discrete series representations, and properly induced representations, when rankg = rankk.
Thank you for attention!