Explanation for Satake correspondence - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T02:17:59Z http://mathoverflow.net/feeds/question/4630 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4630/explanation-for-satake-correspondence Explanation for Satake correspondence Ilya Nikokoshev 2009-11-08T13:32:33Z 2009-11-08T17:41:32Z <p>Some time ago I was told there's an interesting classical Satake correspondence which I will write as </p> <p>$$[\mathop{\mathrm{disk}} \Rightarrow G] \,\backslash\, [\mathop{\mathrm{disk}^\times} \Rightarrow G] \,/\, [\mathop{\mathrm{disk}} \Rightarrow G] \,=\, X_*/W \,=\, G^\vee\mbox{-reps}$$</p> <p>(where $G$ is reductive, $\mathrm{disk}$ could be the spectrum of either $\mathbb Z_p$ or $k[[t]]$ and $\Rightarrow$ denotes algebraic morphism) and its categorified/geometric version (equivariant perverse sheaves on affine grassmanian of $G$ form the tensor category of representations of $G^\vee$).</p> <p>I think I'm missing the larger context here, though. I don't mean the context of perverse sheaves, geometric Langlands, etc. On the contrary, I feel like I miss any intuition for classical representation theory. Why would statements like this be interesting?</p> <p>I wasn't able to find anything in <a href="http://en.wikipedia.org/wiki/Satake%5Fcorrespondence" rel="nofollow">wikipedia</a> or <a href="http://ncatlab.org/nlab/search?%5Fform%5Fkey=4209aaac3878c2dab6d077f12cd5c0363e94a24c&amp;query=satake+correspondence" rel="nofollow">nLab</a>.</p> <p>One thing I know is that the correspondence allows us to construct the Langlands dual group in a natural way. But still, it would be interesting to know if it's a part of larger picture and if there are related results. </p> <blockquote> <p><strong>Question:</strong> is there an intuition for Satake correspondence that would make its statement obvious?</p> </blockquote> http://mathoverflow.net/questions/4630/explanation-for-satake-correspondence/4639#4639 Answer by Ben Webster for Explanation for Satake correspondence Ben Webster 2009-11-08T15:13:18Z 2009-11-08T15:13:18Z <p>What you have written above isn't classical Satake; it's the generalized Bruhat decomposition. Classical Satake is a much more interesting theorem, which says that the Hecke algebra of $G(\mathcal{K})$ over $G(\mathcal{O})$ (the compactly supported $G(\mathcal{O})$ bi-invariant functions on $G(\mathcal{K})$ with convolution multiplication) is isomorphic to the representation <em>ring</em> of $G^\vee$. </p> <p>Why is this interesting? Because the Hecke algebra $L^2[G(\mathcal{O})\backslash G(\mathcal{K})/G(\mathcal{O})]$ is the endomorphism algebra of $L^2(G(\mathcal{K})/G(\mathcal{O}))$, so there's a bijection between $W$-orbits on $T^\vee$ and representations of $G(\mathcal{K})$ appearing in the $L^2$ above.</p>