Namespaces
Variants
Actions

Difference between revisions of "Kirillov conjecture"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX done)
m (typo)
 
Line 8: Line 8:
 
A related conjecture is that for two irreducible representations $\pi_1$ and $\pi_2$ of, respectively, $\mathrm{GL}_{n_1}(F)$ and $\mathrm{GL}_{n_2}(F)$, the product
 
A related conjecture is that for two irreducible representations $\pi_1$ and $\pi_2$ of, respectively, $\mathrm{GL}_{n_1}(F)$ and $\mathrm{GL}_{n_2}(F)$, the product
 
$$
 
$$
\pi_1 \pi_2 \ \mathrm{Ind}_{\mathrm{GL}(n_1,F)\times\mathrm{GL}(n_2,F)}^{\mathrm{GL}(n_1+n_2,F)}
+
\pi_1 \pi_2 = \mathrm{Ind}_{\mathrm{GL}(n_1,F)\times\mathrm{GL}(n_2,F)}^{\mathrm{GL}(n_1+n_2,F)}
 
$$
 
$$
 
is irreducible.
 
is irreducible.

Latest revision as of 08:10, 23 March 2018

Let $F$ be a local field and $\pi$ an irreducible unitary representation of $\mathrm{GL}_n(F)$. Let $$ P_n(F) = \{ s \in \mathrm{GL}_n(F) : \text{last row}\,(s) = (0,0,\ldots,1) \} \ . $$

Then $\pi(P_n(F))$ is irreducible (cf. also Irreducible representation).

A related conjecture is that for two irreducible representations $\pi_1$ and $\pi_2$ of, respectively, $\mathrm{GL}_{n_1}(F)$ and $\mathrm{GL}_{n_2}(F)$, the product $$ \pi_1 \pi_2 = \mathrm{Ind}_{\mathrm{GL}(n_1,F)\times\mathrm{GL}(n_2,F)}^{\mathrm{GL}(n_1+n_2,F)} $$ is irreducible.

For $F$ non-Archimedean (cf. also Archimedean axiom), both conjectures are true (Bernstein's theorems).

For $F = \mathbf{C}$, these conjectures have been proved by S. Sahi [a1].

References

[a1] S. Sahi, "On Kirillov's conjecture for Archimedean fields" Compositio Math. , 72 : 1 (1989) pp. 67–86 Zbl 0693.22006
How to Cite This Entry:
Kirillov conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kirillov_conjecture&oldid=43010
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article