Gelfand Pairs A. Aizenbud and D. Gourevitch www.wisdom.weizmann.ac.il/~aizenr www.wisdom.weizmann.ac.il/~dimagur the compact case the non compact case In the non compact case we consider complex smooth (admissible) representations of algebraic reductive (e.g. GL , O Sp ) groups over local fields (e.g. R, Q ). Fourier Series Spherical Harmonics n n, n p Gelfand Pairs H0 A pair of groups ( G ⊃ H ) is called a Gelfand pair if for any irreducible (admissible) representation ρ of G H ~ 1 dim Hom H (ρ,C) ⋅dim Hom H (ρ,C) ≤ 1. H For most pairs, this implies that 2 dim Hom (ρ,C) ≤ 1. H H 3 Gelfand-Kazhdan Distributional Criterion σ H 4 Let be an involutive anti-automorphism of G and assume σ (H ) = H.

2 Suppose that σ ( ξ ) = ξ for all bi H - invariant distributions 2 1 S = O / O L (S ) = Span (χ ) 3 2 ξa on G. ⊕ m L2 (S 2 ) H m = ⊕ m Then ( G , H ) is a Gelfand pair. (t) eimt m χm = i An analogous criterion works for strong Gelfand pairs H m = Span (Yn ) im α π (α)χm = e χm are irreducibl e

representa tions of O3 Gelfand Pairs A pair of compact topological groups G ⊃ H is called a Gelfand pair if the following equivalent conditions hold: L2 (G / H )decomposes to direct sum of distinct irreducible representations of G. for any irreducible representation ρ of G, dim ρ H ≤ 1 ρ for any irreducible representation of G, dim Hom (ρ |H ,C) ≤1 Results Gelfand pairs Strong Gelfand pairs the algebra of bi-H -invariant functions on G , C ( H \ G / H ) is commutative w.r.t. convolution. Strong Gelfand Pairs A pair of compact topological groups ( G ⊃ H ) is called a strong Gelfand pair if the following equivalent conditions hold: the pair ( G × H ⊃ H ) is a Gelfand pair. Example σ for any irreducible representations ρ of G and τ of H, dim Hom (ρ |H ,τ ) ≤ .1 Any F x - invariant distribution on the plain F 2 is invariant the algebra of Ad ( H ) - invariant functions on G is commutative w.r.t. convolution. with respect to the flip σ.σ.σ. This example implies that (GL , GL ) is a strong Gelfand pair . Gelfand Trick 2 1 2 More generally, Let σ be an involutive anti-automorphism of G (i.e. σ ( g 1 g 2 ) = σ ( g 2 ) σ ( g 1 ) and σ = Id) and assume σ (H ) = H. Any distribution on GL which is invariant w.r.t. Suppose that σ ( f ) = f for all bi-H -invariant functions f ∈C(H \ G / H ). Then ( G , H ) is a Gelfand pair. n+1 conjugation by GL n is invariant w.r.t. transposition . An analogous criterion works for strong Gelfand pairs. −1 This implies that (GL n+1 , GL n) is a strong Gelfand pair . σ (x, y) = (y, x) λ(x, y) = (λx,λ y) Tools to Work with Invariant Distributions Analysis Geometry Integration of distributions – Frobenius Descent Geometric Invariant Theory Luna Slice Theorem −1 X a = f (a) X Classical Examples Classical Applications X U Ga a Gelfand-Zeitlin basis: f p − 1 (a ) G (S ,S ) is a strong Gelfand pair  n n-1 Z = Ga basis for irreducible representations of S n. The same a for O(n, R) and U(n, R). S* G S* Ga Classification of representations: (X) = (X a ) (GL(n, R),O(n, R)) is a Gelfand pair  Fourier transform – uncertainty principle the irreducible representations of GL(n, R) which Algebra have an O(n, R) - invariant vector are the same as D – modules characters of the algebra F Weil representation C(O(n, R)\GL(n, R)/O(n, R)). Representations of SL 2 The same for the pair (GL(n, C),U(n)). F (δ 0 ) =1

Wave front set

Symmetric Pairs

θ We call the property (2) regularity. We conjecture that all symmetric pairs are regular. This will imply the
A pair ( G ⊃ H ) is called a symmetric pair if H = G for some involution θ . conjecture that every good symmetric pair is a Gelfand pair.
We de\note σ (g :) = θ (g −1). Regular pairs Question: What symmetric pairs are Gelfand pairs?
We call a symmetric pair ( G , H , θ ) good if σ preserves all closed H double cosets. Any connected symmetric pair over C is good. Conjecture: Any good symmetric pair is a Gelfand pair. Conjecture: Any symmetric pair over C is a Gelfand pair. How to check that a symmetric pair is a Gelfand pair? 1. Prove that it is good 2. Prove that any H -invariant distribution on g σ is σ -invariant provided that this holds outside the cone of nilpotent elements. 3. Compute all the "descendants" of the pair and prove (2) for them.