Multiplicity one theorems

1
Multiplicity one theorems
A. Aizenbud, D. Gourevitch S. Rallis and G.
Schiffmann
arXiv0709.4215 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
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
2

• 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. Let be a finite G-invariant
stratification. Suppose that for any
i, S(Si)G0. Then S(X)G0.
3
Localization principle
4
Frobenius reciprocity
5
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.

6
Geometric Symmetries
7
Fourier transform
Homogeneity lemma
The proof of this lemma uses Weil representation.
8
(No Transcript)
9
Fourier transform Homogeneity lemma
10
Let D be either F or a quadratic extension of F.
Let V be a vector space over D of dimension n.
Let lt , gt be a non-degenerate hermitian form on
V. Let WV?D. Extend lt , gt 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