Satake isomorphism explained
In mathematics, the Satake isomorphism, introduced by, identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by .
Statement
Classical Satake isomorphism.Let
be a
semisimple algebraic group,
be a non-Archimedean local field and
be its ring of integers. It's easy to see that
is a
grassmannian. For simplicity, we can think that
and
, for
a prime number; in this case,
is an infinite dimensional
algebraic variety . One denotes the category of all compactly supported
spherical functions on
biinvariant under the action of
as
\Complexc[G(O)\backslashG(K)/G(O)]
,
the field of complex numbers, which is a
Hecke algebra and can be also treated as a
group scheme over
. Let
be the maximal torus of
,
be the
Weyl group of
. One can associate a cocharacter variety
to
. Let
be the set of all cocharacters of
, i.e.
X*(T(\Complex))=Hom(\Complex*,T(\Complex))
. The cocharacter variety
is basically the
group scheme created by adding the elements of
as variables to
, i.e.
X*(T(\Complex))=\Complex[X*(T(\Complex))]
. There is a natural action of
on the cocharacter variety
, induced by the natural action of
on
. Then the Satake isomorphism is an algebra isomorphism from the category of
spherical functions to the
-invariant part of the aforementioned cocharacter variety. In formulas:
\Complexc[G(O)\backslashG(K)/G(O)] \xrightarrow{\sim}
.
Geometric Satake isomorphism.As Ginzburg said, "geometric" stands for sheaf theoretic. In order to obtain the geometric version of Satake isomorphism, one has to change the left part of the isomorphism, using the Grothendieck group of the category of perverse sheaves on
to replace the category of
spherical functions; the replacement is de facto an algebra isomorphism over
. One has also to replace the right hand side of the isomorphism by the
Grothendieck group of finite dimensional complex representations of the
Langlands dual
of
; the replacement is also an algebra isomorphism over
. Let
denote the category of
perverse sheaves on
. Then, the geometric Satake isomorphism is
K(Perv(Gr)) ⊗ \Z\Complex \xrightarrow{\sim} K(Rep({}LG)) ⊗ \Z\Complex
,
where the
in
stands for the
Grothendieck group. This can be obviously simplified to
Perv(Gr) \xrightarrow{\sim} Rep({}LG)
,
which is a fortiori an equivalence of Tannakian categories .
References
- Ginzburg . Victor . Perverse sheaves on a loop group and Langlands' duality . alg-geom/9511007 . 2000 .
