description of an endomorphism algebra - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:01:04Z http://mathoverflow.net/feeds/question/99750 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99750/description-of-an-endomorphism-algebra description of an endomorphism algebra th.ng 2012-06-15T22:21:05Z 2012-06-18T00:23:05Z <p>Let $G$ be a reductive group, $F$ a Frobenius morphism, $B$ a Borel subgroup $F$-stable and consider the finite groups $G^F$ and $U^F$ where $U$ is the radical unipotent of $B=UT$ ($T$ torus).</p> <p>I would like a reference for the description of the algebra $End_{G^F}( \mathbb{C}[G^F/U^F] )$. More precisely, I'd like to relate it with a structure of Hecke algebra, which is usually defined as $End_{G^F}( \mathbb{C}[G^F/B^F] ) := End_{G^F} ( Ind_{B^F}^{G^F} 1 )$. I hope to find that the endomorphism algebra is isomorphic to some kind of extension of the Hecke algebra by the torus $T$.</p> <p>Thank you!</p> http://mathoverflow.net/questions/99750/description-of-an-endomorphism-algebra/99846#99846 Answer by Jim Humphreys for description of an endomorphism algebra Jim Humphreys 2012-06-17T16:24:50Z 2012-06-17T16:24:50Z <p>Here you are working over <code>$\mathbb{C}$</code> (or perhaps any other splitting field of characteristic 0 for <code>$G$</code>). So the representation you are starting with is just the direct sum over all characters <code>$\chi$</code> of <code>$T^F$</code> of the various induced characters from <code>$B^F$</code> to <code>$G^F$</code> obtained by lifting <code>$\chi$</code> first to a character of <code>$B^F$</code> and then inducing. All of these induced characters of <code>$G^F$</code> have the same degree, but some are irreducible and others not (as in the extreme case <code>$\chi =1$</code>). So the resulting endomorphism algebra of the large direct sum will be cumbersome to study. It's helpful to consult Chapter 10 of Roger Carter's 1985 book for a more precisely organized program along these lines, due largely to Howlett and Lehrer. Naturally the usual Hecke algebra for <code>$\chi=1$</code> plays a role here, as do analogous endomorphism algebras for other <code>$\chi$</code>. </p> <p>But your expressed hope seems too loosely formulated in this extremely complicated situation. Have you tried to work this out explicitly when <code>$G^F = \mathrm{SL}(2,p)$</code>? In that case all the induced representations are easily identified. </p> <p>By the way, there is a version of all this worked out in the defining characteristic <code>$p$</code> by Carter and Lusztig in their old paper <em>Modular representations of finite groups of Lie type</em>, Proc. London Math. Soc. (3) 32 (1976), no. 2, 347–384. They use BN-pairs as a framework and develop intertwining operators in the spirit of Hecke algebras, but with some degeneracy.</p> http://mathoverflow.net/questions/99750/description-of-an-endomorphism-algebra/99865#99865 Answer by David Ben-Zvi for description of an endomorphism algebra David Ben-Zvi 2012-06-18T00:23:05Z 2012-06-18T00:23:05Z <p>I think Thiem's thesis <a href="http://arxiv.org/abs/math/0402383" rel="nofollow">Unipotent Hecke algebras of GL_n(F_q)</a> discusses this in detail -- if I'm not mistaken the Hecke algebra you're asking about goes by the name Yokonuma Hecke algebra and there's a fair amount of literature on it.</p>