A. Aizenbud and D. Gourevitch www.wisdom.weizmann.ac.il/~aizenr www.wisdom.weizmann.ac.il/~dimagur Gelfand Pairs Regular pairs We call the property (2) regularity. We conjecture that all symmetric pairs are regular. This will imply the conjecture that every good symmetric pair is a Gelfand pair. A pair is called a symmetric pair if for some involution We de\note Question: What symmetric pairs are Gelfand pairs? We call a symmetric pair good if preserves all closed 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 -invariant distribution on is -invariant provided that this holds outside the cone of nilpotent elements. 3.Compute all the "descendants" of the pair and prove (2) for them. Symmetric Pairs H ) , , ( H G H g A pair of compact topological groups is called a Gelfand pair if the following equivalent conditions hold: decomposes to direct sum of distinct irreducible representations of for any irreducible representation of for any irreducible representation of the algebra of bi- -invariant functions on is commutative w.r.t. convolution. Gelfand Pairs Fourier Series Spherical Harmonics ) ( ) ( 1 2 m m Span S L m m H S L ) ( 2 2 imt m e t ) ( m im m e ) ( ) ( i n m Y Span H 3 O of tions representa e irreducibl are 0 H 1 H 2 H 4 H 3 H 2 3 2 / O O S Strong Gelfand Pairs A pair of compact topological groups is called a strong Gelfand pair if the following equivalent conditions hold: the pair is a Gelfand pair. for any irreducible representations the algebra of - invariant functions on is commutative w.r.t. convolution. the compact case Classical Applications Gelfand-Zeitlin basis: (S n ,S n-1 ) is a strong Gelfand pair basis for irreducible representations of S n . The same for O(n,R) and U(n,R). Classification of representations: (GL(n,R),O(n,R)) is a Gelfand pair the irreducible representations of GL(n,R) which have an O(n,R) - invariant vector are the same as characters of the algebra C(O(n,R)\GL(n,R)/O(n,R)). The same for the pair (GL(n, C),U(n)). Classical Examples ) / ( 2 H G L . G H 1 ) , | ( Hom dim , C H G G 1 dim , H G ) / \ ( , H G H C G ) ( H G . 1 ) , | ( Hom dim , of and of H H G ) ( Ad H H G ) ( H H G Gelfand Trick Let be an involutive anti-automorphism of and assume Suppose that for all bi- -invariant functions Then is a Gelfand pair. An analogous criterion works for strong Gelfand pairs. G H . ) ( H H Id) and ) ( ) ( ) ( (i.e. 2 1 2 2 1 g g g g f f ) ( ). / \ ( H G H C f ) , ( H G G H . ) ( H G ). ( : ) ( 1 g g the non compact case In the non compact case we consider complex smooth (admissible) representations of algebraic reductive (e.g. GL n , O n, Sp n ) groups over local fields (e.g. R, Q p ). Result s Example Tools to Work with Invariant Distributions Gelfand pairs Strong Gelfand pairs Any F x - invariant distribution on the plain F 2 is invariant with respect to the flip This example implies that (GL 2 , GL 1 ) is a strong Gelfand pair. More generally, Any distribution on GL n+1 which is invariant w.r.t. conjugation by GL n is invariant w.r.t. transposition. This implies that (GL n+1 , GL n ) is a strong Gelfand pair. Analysis Integration of distributions – Frobenius Descent Fourier transform – uncertainty principle Wave front set a G a * G * ) (X (X) S S Ga Z ) ( 1 a f X a X f a G F Algebra D – modules Weil representation Representations of SL 2 Geometry Geometric Invariant Theory Luna Slice Theorem X U Ga a ) ( 1 a p Gelfand Pairs Gelfand-Kazhdan Distributional Criterion A pair of groups is called a Gelfand pair if for any irreducible (admissible) representation of For most pairs, this implies that ) ( H G G 1. ) , ~ ( dim ) , ( dim C C H H Hom Hom 1. ) , ( dim C H Hom Let be an involutive anti-automorphism of and assume Suppose that for all bi - invariant distributions a on Then is a Gelfand pair. An analogous criterion works for strong Gelfand pairs G . ) ( H H ) ( H . G ) , ( H G ) , ( ) , ( x y y x ) , ( ) , ( 1 y x y x 1 ) ( 0 F