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Multiplicity one theorems. (English) Zbl 1202.22012
Consider the standard upper left hand corner embedding of $$\mathrm{GL}_n(F)$$ in $$\mathrm{GL}_{n+1}(F)$$ where $$F$$ is a non-Archimedean local field of characteristic $$0$$. The main result is the following “multiplicity $$\leq 1$$” theorem and its analogues for orthogonal and unitary groups:
If $$\pi_n, \pi_{n+1}$$ are irreducible, admissible representations of $$\mathrm{GL}_n(F), \mathrm{GL}_{n+1}(F)$$ respectively, then $$\dim (\operatorname{Hom}_{\mathrm{GL}_n(F)}(\pi_{n+1}|\mathrm{GL}_n(F), \pi_n) ) \leq 1$$.
Analogues of this theorem for Archimedean $$F$$ were obtained in special cases earlier. “Multiplicity $$\leq 1$$” theorems have several applications to the study of automorphic $$L$$-functions. But a more difficult question is to find when the multiplicity is 1. Recently, the above theorem has been used to deduce the analogous result when $$F$$ has positive characteristic.
The transposition map on $$\mathrm{GL}_{n+1}(F)$$ is an anti-automorphism of order $$2$$ which leaves $$\mathrm{GL}_{n}(F)$$ stable. The above theorem is deduced from: “Consider the (adjoint) action of $$\mathrm{GL}_n(F) \times \mathrm{GL}_n(F)$$ on $$\mathrm{GL}_{n+1}(F)$$ given by $$(g_1,g_2)h = g_1hg_2^{-1}$$. Then, any invariant distribution on $$\mathrm{GL}_{n+1}(F)$$ with respect to this action is also invariant with respect to transposition.”
Employing an old result due to Bernstein, the above theorem also provides an independent proof of Kirillov’s conjecture for non-Archimedean fields of characteristic $$0$$. The analogues of the two stated theorems are proved for unitary groups also. Recently, Waldspurger has adapted the proofs to include special orthogonal groups. The proof of the above theorem uses the powerful Bernstein localization principle and a variant of Frobenius reciprocity from 1984. The authors also point out that there is still no simple explanation why the invariant distributions always turn out to be symmetric.

##### MSC:
 22E35 Analysis on $$p$$-adic Lie groups 20G25 Linear algebraic groups over local fields and their integers
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