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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101101.png" /> be a [[Local field|local field]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101102.png" /> an irreducible [[Unitary representation|unitary representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101103.png" />. Let
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Let $F$ be a [[local field]] and $\pi$ an irreducible [[unitary representation]] of $\mathrm{GL}_n(F)$. Let
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$$
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P_n(F) = \{ s \in \mathrm{GL}_n(F) : \text{last row}\,(s) = (0,0,\ldots,1) \} \ .
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$$
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101104.png" /></td> </tr></table>
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Then $\pi(P_n(F))$ is irreducible (cf. also [[Irreducible representation]]).
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Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101105.png" /> is irreducible (cf. also [[Irreducible representation|Irreducible representation]]).
 
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A related conjecture is that for two irreducible representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101107.png" /> of, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k1101109.png" />, the product
 
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k11011010.png" /></td> </tr></table>
 
  
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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
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$$
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\pi_1 \pi_2 = \mathrm{Ind}_{\mathrm{GL}(n_1,F)\times\mathrm{GL}(n_2,F)}^{\mathrm{GL}(n_1+n_2,F)}
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$$
 
is irreducible.
 
is irreducible.
  
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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k11011011.png" /> non-Archimedean (cf. also [[Archimedean axiom|Archimedean axiom]]), both conjectures are true (Bernstein's theorems).
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For $F$ non-Archimedean (cf. also [[Archimedean axiom]]), both conjectures are true (Bernstein's theorems).
  
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For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110110/k11011012.png" />, these conjectures have been proved by S. Sahi [[#References|[a1]]].
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For $F = \mathbf{C}$, these conjectures have been proved by S. Sahi [[#References|[a1]]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Sahi,  "On Kirillov's conjecture for Archimedean fields"  ''Compositio Math.'' , '''72''' :  1  (1989)  pp. 67–86</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Sahi,  "On Kirillov's conjecture for Archimedean fields"  ''Compositio Math.'' , '''72''' :  1  (1989)  pp. 67–86 {{ZBL|0693.22006}}</TD></TR>
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</table>
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{{TEX|done}}

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=14218
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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