Quantum groups, the loop Grassmannian, and the Springer resolution
Authors: Arkhipov Sergey , Bezrukavnikov Roman , Ginzburg Victor 
Published: May 2004

We establish equivalences of the following three triangulated categories:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Subscript quantum Baseline left-parenthesis German g right-parenthesis long left-right-arrow upper D Subscript coherent Superscript upper G Baseline left-parenthesis script upper N overTilde right-parenthesis long left-right-arrow upper D Subscript perverse Baseline left-parenthesis sans-serif upper G sans-serif r right-parenthesis period">

D
quantum

(

g

)

⟷

D
coherent
G

(

N

~

)

⟷

D
perverse

(

G
r

)
.

D_\text {quantum}(\mathfrak {g})\enspace \longleftrightarrow \enspace D^G_\text {coherent}(\widetilde {{\mathcal N}})\enspace \longleftrightarrow \enspace D_\text {perverse}(\mathsf {Gr}).

\]
Here,

D
quantum

(

g

)

D_\text {quantum}(\mathfrak {g})

is the derived category of the principal block of finite-dimensional representations of the quantized enveloping algebra (at an odd root of unity) of a complex semisimple Lie algebra

g

\mathfrak {g}

; the category

D
coherent
G

(

N

~

)

D^G_\text {coherent}(\widetilde {{\mathcal N}})

is defined in terms of coherent sheaves on the cotangent bundle on the (finite-dimensional) flag manifold for

G
G

(

=
=

semisimple group with Lie algebra

g

\mathfrak {g}

), and the category

D
perverse

(

G
r

)

D_\text {perverse}({\mathsf {Gr}})

is the derived category of perverse sheaves on the Grassmannian

G
r

{\mathsf {Gr}}

associated with the loop group

L

G
∨

LG^\vee

, where

G
∨

G^\vee

is the Langlands dual group, smooth along the Schubert stratification. The equivalence between

D
quantum

(

g

)

D_\text {quantum}(\mathfrak {g})

and

D
coherent
G

(

N

~

)

D^G_\text {coherent}(\widetilde {{\mathcal N}})

is an “enhancement” of the known expression (due to Ginzburg and Kumar) for quantum group cohomology in terms of nilpotent variety. The equivalence between

D
perverse

(

G
r

)

D_\text {perverse}(\mathsf {Gr})

and

D
coherent
G

(

N

~

)

D^G_\text {coherent}(\widetilde {{\mathcal N}})

can be viewed as a “categorification” of the isomorphism between two completely different geometric realizations of the (fundamental polynomial representation of the) affine Hecke algebra that has played a key role in the proof of the Deligne-Langlands-Lusztig conjecture. One realization is in terms of locally constant functions on the flag manifold of a

p
p

-adic reductive group, while the other is in terms of equivariant

K
K

-theory of a complex (Steinberg) variety for the dual group. The composite of the two equivalences above yields an equivalence between abelian categories of quantum group representations and perverse sheaves. A similar equivalence at an even root of unity can be deduced, following the Lusztig program, from earlier deep results of Kazhdan-Lusztig and Kashiwara-Tanisaki. Our approach is independent of these results and is totally different (it does not rely on the representation theory of Kac-Moody algebras). It also gives way to proving Humphreys’ conjectures on tilting

U
q

(

g

)

U_q(\mathfrak {g})

-modules, as will be explained in a separate paper.