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					      Multiplicity one theorems
            A. Aizenbud, D. Gourevitch S. Rallis and G. Schiffmann
                        arXiv:0709.4215 [math.RT]

Let F be a non-archimedean local field of characteristic zero.

Theorem A Every GL(n; F) invariant distribution on GL(n + 1; F) is
invariant with respect to transposition.
         it implies

Theorem B Let p be an irreducible smooth representation of GL(n + 1; F)
and let r be an irreducible smooth representation of GL(n; F). Then

             dim HomGL( n, F ) (p    GL( n , F )   , r)  1

Theorem B2 Let p be an irreducible smooth representation of O(n + 1; F)
and let r be an irreducible smooth representation of O(n; F). Then

               dim HomO( n, F ) (p    O( n, F )    , r)  1
• Let X be an l-space (i.e. Hausdorff locally compact totally
  disconnected topological space). Denote by S(X) the
  space of locally constant compactly supported functions.
• Denote also S*(X):=(S(X))*
• For closed subset Z of X,
  0 → S*(Z) → S*(X) → S*(X\Z) →0.
  Corollary. Let an l-group G act on an l-space X.
             n
  Let   X   Si   be a finite G-invariant stratification. Suppose that for any i,
            i 1


  S*(Si)G=0. Then S*(X)G=0.
Localization principle
Frobenius reciprocity
    Proof of Gelfand-Kazhdan Theorem
Theorem (Gelfand-Kazhdan). Every GL(n ; F) invariant
distribution on GL(n ; F) is invariant with respect to transposition.

Proof
• Reformulation:



• Localization principle
                     Here, q is the “characteristic polynomial” map,
                     and P is the space of monic polynomials of degree n.

• Every fiber has finite number of orbits
• For every orbit we use Frobenius reciprocity and the fact
  that A and At are conjugate.
Geometric Symmetries
                Fourier transform



                  Homogeneity lemma




The proof of this lemma uses Weil representation.
        Tool              Used in            Used by
Bernstein’s
localization                               Gel’fand and
principle                (GLn , GLn)        Kazhdan
Frobenius
reciprocity
Fourier transform        (Pn , GLn)         Bernstein

                      (GLnxGLk , GLn+k)    Jacquet and
Weil representation     two-sided action      Rallis

                                            Rallis and
                        (GLn , GLn+1)       Schiffmann
Geometric                                  Aizenbud and
Symmetries                                  Gourevitch
Fourier transform &
Homogeneity lemma
Let D be either F or a quadratic extension of F. Let V be a vector space
over D of dimension n. Let < , > be a non-degenerate hermitian form on
V. Let W:=V⊕D. Extend < , > to W in the obvious way. Consider the
embedding of U(V) into U(W).

Theorem A2 Every U(V)- invariant distribution on U(W) is invariant with
respect to transposition.
         it implies

Theorem B2 Let p be an irreducible smooth representation of U(W)
and let r be an irreducible smooth representation of U(V). Then

                  dim HomU (V ) (p   U (V )   , r)  1

				
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