a question about irreducibility of representations and Kirillov conjecture - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T00:03:13Z http://mathoverflow.net/feeds/question/19788 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19788/a-question-about-irreducibility-of-representations-and-kirillov-conjecture a question about irreducibility of representations and Kirillov conjecture unknown (google) 2010-03-30T02:38:33Z 2010-03-30T19:41:29Z <p>Let $G=GL(\mathbb{R})$, $P$ be the subgroup of $G$ consisting of elements with the last row $(0,0,...,1)$. Then Kirillov conjecture states that for any irreducible unitary representation of $G$, its restriction to $P$ remains irreducible. This conjecture has been proved (not only over $\mathbb{R}$, but also over $\mathbb{C}$ and p-adic fields). Here I'm wondering if we consider irreducible smooth representations in Hilbert space(or Banach, Frechet space), does this conjecture remains true?</p> <p>Another related question is generally, how to prove the irreducibility for a smooth representation besides the definition? </p> http://mathoverflow.net/questions/19788/a-question-about-irreducibility-of-representations-and-kirillov-conjecture/19844#19844 Answer by Marty for a question about irreducibility of representations and Kirillov conjecture Marty 2010-03-30T15:52:31Z 2010-03-30T19:41:29Z <p>I think it's best to look at the relatively recent paper of Moshe Baruch, Annals of Math., "A Proof of Kirillov's Conjecture" -- in the introduction of his paper, he discusses the basic techniques of proof, and a bit of the history (Bernstein proved this conjecture in the p-adic case, for example).</p> <p>Baruch and others (e.g. Kirillov, in the original conjecture, I think) consider unitary representations. This is necessary for the methods which they use. From the beginning, they use the "converse of Schur's lemma", i.e., if $Hom_P(V,V)$ is one-dimensional then $V$ is irreducible. This converse of Schur's lemma requires one to work in the unitary (or unitarizable) setting.</p> <p>Now, to address Kevin Buzzard's point, consider $G = GL_2(F)$ for a $p$-adic field, and a unitary principal series representation $V = Ind_B^G \chi \delta^{1/2}$, where $\chi = \chi_1 \boxtimes \chi_2$ is a unitary character of the standard maximal torus, and $\delta$ is the modular character for the Borel $B$.</p> <p>Restricting $V$ back down to $B$, one gets a short exact sequence of $B$-modules: $$0 \rightarrow V(BwB) \rightarrow V \rightarrow V(B) \rightarrow 0,$$ where $V(X)$ denotes a space of functions (compactly supported modulo $B$ on the left) on the ($B$-stable) locus $X$. On can check that these spaces are nonzero using the structure of the Bruhat cells, and hence the restriction of $V$ to $B$ is reducible as Kevin suggests. </p> <p><em>But</em>, if one considers the Hilbert space completion $\hat V$ of $V$, with respect to a natural Hermitian inner product, one finds that $\hat V$ is an irreducible unitary representation of $G$ which remains irreducible upon restriction to $B$ (and to the even smaller "mirabolic" subgroup of Kirillov's conjecture). Here it is important to note that "irreducibility" for unitary representations on Hilbert spaces refers to <em>closed</em> subspaces. The $B$-stable subspace $V(BwB)$ of $V$ is not closed, and its closure is all of $\hat V$ I think.</p> <p>So - I think that Kirillov's conjecture is false, in the setting of smooth representations of $p$-adic groups (and most probably for smooth representations of moderate growth of real groups).</p> <p>However, the techniques still apply in the smooth setting to give weaker (but still useful) results. After all, it is still useful to know that $Hom_P(V,V)$ is one-dimensional! This can be used to prove multiplicity one for certain representations, for example.</p> <p>The general technique to prove $Hom_P(V,V)$ is one-dimensional involves various forms of Frobenius reciprocity and characterization of distributions. Without explaining too much (you should look at old papers of Bernstein, perhaps), and being sloppy about dualities sometimes, $$Hom_P(V,V) \cong Hom_P(End(V), C) \cong Hom_G(End(V), Ind_P^G C).$$ Some sequence of Frobenius reciprocity and linear algebra (I don't think I have it quite right above) identifies $Hom_P(V,V)$ with a space of functions or distributions: $f: G \rightarrow End(V)$, which are $(P,V)$-bi-invariant. In other words, $$f(p_1 g p_2) = \pi(p_1^{-1}) \circ f(g) \circ \pi(p_2),$$ or something close. </p> <p>So in the end, one is led to classify a family of $P$-bi-quasi-invariant $End(V)$-valued distributions on $G$. This leads to two problems: one geometric, involving the $P$-double cosets in $G$. This is particularly easy for the "mirabolic" subgroup $P$. The second problem is often more difficult, analyzing distributions on each double coset, and proving most of them are zero or else have very simple properties. </p> <p>Hope this clarifies a little bit... you might read more on the Gelfand-Kazhdan method (Gross has an exposition in the Bulletin) to understand this better.</p>