Let $G$ be a complex reductive group, $L\subset G$ a Levi subgroup and $Rep(G)$ the category of rational representations of $G$.

My Question: What is the geometric analogue of the restriction functor $Res^G_L:Rep(G)\to Rep(L)$?

To be a little bit more precise:

Let $\check{G}$ be the dual group of $G$. As usual define $\mathcal{K}:=\mathbb{C}$ and $\mathcal{O}:=\mathbb{C}[[t]]$. Let further $Gr_{\check{G}}:=\check{G}(\mathcal{K})/\check{G}(\mathcal{O})$ denote the affine Grassmannian and $P_{\check{G}(\mathcal{O})}(Gr_{\check{G}})$ the category of $\check{G}(\mathcal{O})$-equivariant perverse sheaves on $Gr$. The geometric Satake Isomorphism gives an equivalence of tensor categories categories $$P_{\check{G}(\mathcal{O})}(Gr_{\check{G}}) \cong Rep(G)$$

Is there a nice (from the geometric viewpoint) functor $\check{Res}:P_{\check{G}(\mathcal{O})}(Gr_{\check{G}})\to P_{\check{L}(\mathcal{O})}(Gr_{\check{L}})$ which corresponds under the geometric Satake isomorphism to the restriction functor $Res^G_L$?

  • $\begingroup$ On the level of functions it is the "constant term map" , see e.g. math.ucdavis.edu/~kapovich/EPR/HKM.pdf Somebody probably worked this out on the level of sheaves too. $\endgroup$ – Misha Sep 20 '13 at 12:23
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    $\begingroup$ I can't find a reference, but I would guess that some pull-push formula like math.harvard.edu/~gaitsgde/GL/CT.pdf should work. $\endgroup$ – Clemens Koppensteiner Sep 20 '13 at 15:02
  • $\begingroup$ To give one perspective on your first question (although not on the more precise formulation), the flag variety of $L$ is a natural subvariety of the flag variety of $G$, and the restriction functor identifies with restriction of sections of bundles. This gives a geometric interpretation, although it probably doesn't say much about the case of the affine Grassmannian. $\endgroup$ – Chuck Hague Sep 20 '13 at 15:29

On the level of sheaves, the construction is indeed a pull-push formula. I believe it was first worked out by Beilinson and Drinfeld in section 5.3 of their preprint "Quantization of Hitchin's Hamiltonians and Hecke eigen-sheaves". For published references, look at section 2.4 in Braverman and Gaitsgory's paper "Crystals via the affine Grassmannian" (Duke Math. J. 107 (2001), 561–575) or at section 1.3 in Vasserot's paper "On the action of the dual group on the cohomology of perverse sheaves on the affine Grassmannian" (Compositio Math. 131 (2002), 51–60).

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    $\begingroup$ Let me make one comment about it. Although, the construction is indeed as described above, there is one subtle thing about it. Namely, on the representation theory side, the restriction functor depends only on $L$ but not a choice of a parabolic $P$. However, geometrically the functor depends on the parabolic very heavily (note that you can have non-conjugate parabolics with the same Levi). I don't know any geometric way (without using Satake equivalence) to prove independence of $P$ - this is actually a very good problem (but probably it has no reasonable solution). $\endgroup$ – Alexander Braverman Nov 18 '13 at 11:01

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