RESEARCH STATEMENT
VINOTH NANDAKUMAR
Contents
1. Geometry of the exotic nilpotent cone
1
2. Exotic t-structures
3
3. Quiver varieties and the B(∞) crystal
5
References
6
My research focuses on studying the geometry and combinatorics of objects that appear in representationtheoretic contexts. Much of my work is centered on extracting representation theoretic information about
a semi-simple Lie group from the geometry of its nilpotent cone, and other Springer theoretic varieties.
While such varieties were first introduced to study representations of the Weyl group, more recent work
has established links with modular representations of Lie algebras, categorification, affine Hecke algebras
and category O. In particular, I am interested in exotic t-structures on categories of coherent sheaves
on Springer theoretic varieties, which were recently introduced by Bezrukavnikov and Mirkovic to study
representation theory of Lie algebras in positive characteristic; and in the geometry of the exotic nilpotent
cone, which was introduced by Kato as a substitute for the type C nilpotent cone with better properties.
I have also worked with realizing crystal bases for quantum groups using Lusztig’s quiver varieties, using
Kashiwara-Saito’s approach.
In this statement, I will summarize completed work, and outline related projects which I intend to work
on in the future.
1. Geometry of the exotic nilpotent cone
Many aspects of the representation theory of a semisimple Lie group can be seen by studying the geometry
of its nilpotent cone. However, the Springer correspondence in types B, C and D is more complicated
than that in type A in many ways. Kato has introduced the “exotic nilpotent cone”, as a substitute for
the nilpotent cone in type C which mimicks its behaviour and in many cases has better properties.
Let G = Sp2n (C) and g = sp2n (C) be the symplectic group and its Lie algebra (here C2n is equipped
with a symplectic form). Kato’s exotic nilpotent is defined to be N = C2n × N0 where:
N0 = {x ∈ End(C2n ) | x nilpotent, hxv, vi = 0 ∀ v ∈ V}
In [2], Kato shows that the G-orbits on N are in bijection with Qn (the set of bi-partitions of n), and
shows that the top cohomology of the exotic Springer fibers carries a Weyl group action and realizes
the corresponding irreducible representation. Kato’s correspondence is cleaner than the type C Springer
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VINOTH NANDAKUMAR
correspondence, since all local systems on orbits are trivial. In [1], Kato has studied multi-parameter
affine Hecke algebras using the equivariant K-theory of the exotic Steinberg variety (following techniques
used Kazhdan, Lusztig and Ginzburg).
1.1. Previous work. Now let G be any semisimple algebraic group; denote by g its Lie algebra and
N ⊂ g its nilpotent cone. Let Λ+ denote the set of dominant weights for G, and O denote the set of
pairs (O, V ), where O is a G-orbit on N , and V is a finite-dimensional irreducible representation of the
isotropy group Gx of the orbit O, where x ∈ O. Motivated by the theory of two-sided cells in affine Weyl
groups, Lusztig and Vogan conjectured (independently) that there is a canonical bijection between Λ+
and O. Using geometric methods, in [9], Bezrukavnikov proves this conjecture by studying the irreducible
objects in a certain t-structure on Db (CohG (N )). It is also shown in [8] that on the level of Grothendieck
∗
∗
groups, the natural map Db (CohG×C (St)) → Db (CohG×C (N )) sends a canonical basis element in Haf f
either to 0, or to the class of an irreducible object.
Now specialize to G = Sp2n (C); in [18], we follow the method used in [9] to establish an exotic analogue
of this bijection. Let O be the set of pairs (O, L), where O is a G-orbit on N, and L is a finite-dimensional
irreducible representation of the isotropy group Gx of the orbit O (here x ∈ O). In Theorem 4.8 of [18]
we construct a bijection between Λ+ and O.
The quasi-exceptional t-structure on Db (CohG (N)): Let C = Db (CohG (N)). In [1], [2], Kato
e → N; we also have the vector bundle projection
constructs a semi-small resolution of singularities π : N
e → G/B. Let O e (λ) = p∗ OG/B (λ), and denote ∇λ = Rπ∗ O e (λ)[d] where d = dim(N) . In Section 2
p:N
N
N
2
of [18], we describe the cohomology of the sheaves ONe (λ).
Let θ = 21 (1 + 2 + · · · + n ) and define the twisted action w · λ = w(λ + θ) − θ for w ∈ W, λ ∈ Λ (motivated
by the fact that ωNe = ONe (−2θ)). For µ ∈ Λ, let c]
onv(µ) denote the intersection of the convex hull of the
set {w · µ|w ∈ W } in Λ ⊗Z R with Λ. In Theorem 3.16 of [18], we prove that:
Theorem 1.1. There is a unique t-structure on C = Db (CohG (N)), such that ∇λ ∈ C ≥0 and ∇w0 ·λ ∈ C ≤0
for λ ∈ Λ+ .
To show this, we use the theory of a quasi-exceptional sets in a triangulated category C (see Section 2 of
[9] for a detailed discussion). The notion is similar to that of a highest weight category, and consists of
two sets of objects: {∇i |i ∈ I}, and it’s dual {∆i |i ∈ I} (where I is a totally ordered indexed set), which
satisfy certain Hom-vanishing conditions and generate C. It is known that there is a unique t-structure
on C with ∇i ∈ C ≥0 , ∆i ∈ C ≤0 , and that the simple objects in the heart are naturally indexed by I.
Perverse coherent t-structures: Let X be an algebraic variety with an action of an algebraic group G,
with finitely many orbits. In [7], Bezrukavnikov defines a “perverse coherent” t-structure on Db (CohG (X))
(here we are using the middle perversity function). In Corollaries 4 and 5 of [7], it is proven that the
irreducible objects in the heart P of the perverse t-structure are parametrized by a G-orbit O on X, and
an irreducible G-equivariant vector bundle L on O. Applying these results to G = Sp2n (C), X = N, we
have a perverse coherent t-structure defined on Db (CohG (N)). In Proposition 4.6 of [18], we prove that:
Theorem 1.2. The perverse coherent t-structure on Db (CohG (N)) coincides with the quasi-exceptional
t-structure corresponding to the set {∇λ |λ ∈ Λ}.
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The irreducible objects in the heart P of the perverse coherent t-structure are parametrized by O (since
an irreducible G-equivariant vector bundle L on an orbit O is equivalent to giving an irreducible representation of the isotropy group). But using the above results, the irreducible objects in the heart of the
quasi-exceptional t-structure are parametrized by Λ+ . Since the two t-structures coincide, the desired
bijection between Λ+ and O follows.
1.2. Future work.
e (analogous to that constructed
In future work, I plan to construct an ”exotic” t-structure on Db (CohG (N))
∗
∗
by Bezrukavnikov in [6]), give a description of the natural map Db (CohG×C (St)) → Db (CohG×C (N ) on
the level of Grothendieck groups, and investigate the connection with these t-structures (in the classical
case, it is known that the images of the irreducibles in these t-structures are related to canonical bases
in the affine Hecke algebra).
In an separate project, joint with Daniele Rosso and Neil Saunders, I plan to give a geometric construction
of the type C Robinson-Schensted correspondence using the exotic nilpotent cone. Recall that in type A
the Robinson-Schensted correspondence gives an explicit combinatorial bijection (which is motivated by
representation theory of Sn ):
G
SYT(λ) × SYT(λ)
(1)
Sn ←→
λ∈Pn
e ×N N
e be the Steinberg variety (where N
e = T ∗ G/B → N is the
Let G = SLn (C), and let St = N
Springer resolution of its nilpotent cone). By studying the pre-images of certain subvarieties under the
projection maps p : St → G/B × G/B, π : St → N , in [23] Steinberg parametrizes the irreducible
components of St in two different ways, and recovers the above bijection. The standard Young tableau
of shape λ appear as irreducible components of the corresponding Springer fiber (following Spaltenstein’s
result from [21]).
While this bijection can be generalized to arbitrary semi-simple Lie groups, in the other classical types the
bijection differs from the version of the Robinson-Schensted correspondence one expects from studying
representations of the Weyl group. However, by instead studying the geometry of the exotic Steinberg
e N N,
e we expect to recover the type C Robinson-Schensted correspondence. The first step
variety St = N×
is to extend Spaltenstein’s result from [21], and prove that irreducible components of the exotic Springer
fiber for the orbit Oµ,ν are indexed by standard Young tableau for the bi-partition (µ, ν).
2. Exotic t-structures
Let g be a semisimple Lie algebra, and let e
g → g be the Grothendieck map. In [10], Bezrukavnikov and
eS )) in order to study
Mirkovic introduce “exotic t-structures” on the categories Db (Coh(e
gS )), Db (Coh(N
eS denote exact base changes of S). These exotic
the modular representation theory of g (here e
gS and N
t-structures are defined a certain action of the affine braid group Baf f on these categories.
More precisely, let k be an algebraically closed field of characteristic p with p > h (here h is the Coxeter
number). Let λ ∈ hk be integral and regular; and let e ∈ N (k) be a nilpotent. Let Modfe g,λ (Uk ) be
the category of modules with generalized central character (λ, e). In Theorem 5.3.1 from [11] (see also
Section 1.6.2 from [10]), states that there is an equivalence:
Db (CohBe,k (e
gk )) ' Db (Modfe g,λ (Uk ))
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VINOTH NANDAKUMAR
Further, it is proven that the tautological t-structure on the derived category of modules, corresponds to
the exotic t-structure on the derived category of coherent sheaves.
2.1. Previous work. In [5] (joint with Rina Anno), we study exotic t-structures in type A, for nilpotents
whose Jordan type has two blocks; our main result is an explicit description of the irreducible objects
in the heart of the exotic t-structure. So fix m ≥ 0; let n ∈ Z≥0 vary, and let g = slm+2n . Suppose
e → N is the Springer resolution, and let z be the standard nilpotent of type (m + n, n). The
π : N
Springer fiber is the variety
Bz = π −1 (z) = {(0 ⊂ V1 ⊂ · · · ⊂ Vm+2n = Cm+2n ) | zVi ⊆ Vi−1 }
Let S ⊂ g be the Slodowy slice at z; and U its pre-image under the Springer resolution. We will be
studying the categories Dn = Db (CohBz (U )) (the bounded derived category of coherent sheaves on U
supported on Bz ). By the results of [10], these categories are equipped with exotic t-structures. Our first
main result is Theorem 4.26 of [5], where we techniques developed by Cautis and Kamnitzer in [13] to
prove that:
Theorem 2.1. Let ATanm be the category with objects {m + 2k} for k ∈ Z≥0 , and the morphisms
between {m + 2p} and {m + 2q} consist of all affine (m + 2p, m + 2q) tangles (up to isotopy). Here
an affine (m + 2p, m + 2q)-tangle consists of a diagram with m + 2p points on an inner circle, m + 2q
points on an outer circle, arcs connecting these endpoints, and a finite number of circles. Then there is a
weak representation of the category ATanm using the categories Dk (ie. for each affine (m + 2p, m + 2q)
tangle α there is a functor Ψ(α) : Dp → Dq , and an isomorphism Ψ(β) ◦ Ψ(α) ' Ψ(β ◦ α) for each
(m + 2q, m + 2r)-tangle β).
The constructed action of ATanm on the categories Dk is useful because the Baf f -action (which is used
to define exotic t-structures) can be extracted from it. Our main result (Proposition 5.10 in [5]) is that:
Theorem 2.2. Let Cross(n) be the set of affine (m, m + 2n) tangles without crossings, where the m inner
points are not labelled. For each α ∈ Cross(n), we have a functor Ψ(α) : D0 → Dn ; let Ψα = Ψ(α)(C)
(noting that D0 ' Db (Vect)). Then {Ψα | α ∈ Cross(n)} constitute the irreducible objects in Dn0 .
We also give an explicit combinatorial description of Ext• (Ψα , Ψβ ) when α, β ∈ Cross(n) (see Theorem
5.15 in [5]).
2.2. Future work.
CKL functors for Nakajima’s quiver varieties in type A, and exotic t-structures
Let g be a symmetrizable Kac-Moody algebra, and Uq (g) it’s quantum group. Fix w ∈ NI ; but allow
P
P
v ∈ NI to vary. Let Λw = i∈I wi Λi and let λv = Λw − i∈I vi αi . Nakajima constructs quiver varieties
L
×
M(v, w), and constructs an action of Uq (g) on v K C (M(v, w)) (see, for instance, [17]); the resulting
×
representation contains the irreducible with highest weight Λw with multiplicity one, and K C (M(v, w))
corresponds to the weight space λv .
In [14], Cautis, Kamnitzer and Licata categorify this construction, by upgrading the operators Ei , Fi
between weight spaces to functors between the categories:
Ei (v) : Db Coh(M(v, w)) → Db Coh(M(v − αi , w)), Fi (v) : Db Coh(M(v, w)) → Db Coh(M(v + αi , w))
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Now let g = sln , and let w = kΛ1 . It is known (see Lemma 5 from [16]) that M(v, w) is empty unless
k ≥ v1 ≥ · · · ≥ vn−1 , in which case M(v, w) ' T ∗ Gr(vn−1 , · · · , v1 , k) (here Gr(vn−1 , · · · , v1 , k) denotes
flags of type (vn−1 , · · · , v1 ) inside an ambient space of dimension k). Let G0 = GL(k), and P 0 be the
parabolic such that G0 /P 0 = Gr(vn−1 , · · · , v1 , k). Using the results of Bezrukavnikov and Mirkovic from
Sections 3 and 4 of [10], the categories Db (Coh(T ∗ G0 /P 0 )) are equipped with exotic t-structures.
In a future project, I plan to show that the functors Ei (v) are Koszul dual to exact functors with respect
to the exotic t-structures on the domain and target. Now the irreducible objects in the heart of the exotic
t-structure will provide a distinguished basis in the corresponding representation. Knowing the exactness
of the functors Ei (v) will imply the positivity of the structure constants of the operators Ei with respect
to this basis. It will also provide an abelian categorification of the corresponding representation of Uq (g).
Stability conditions for category O.
Inspired by Bridgeland’s theory of stability conditions, in [4], Anno, Bezrukavnikov and Mirkovic define
the notion of a “real variation of stability conditions” on a triangulated category. They then give an
example of this construction, using exotic t-structures on the derived category of coherent sheaves on a
Springer fiber.
We briefly recall their construction (see Section 1.4 of [4] for more details). Let C be a triangulated
category, and Σ be a discrete collection of affine hyperplanes in a finite-dimensional, real vector space V .
Let Alc be the set of connected components of V 0 = V \Σ. Then an real variation of stability conditions
on C parametrized by V 0 consists of a polynomial map Z : V → (K 0 (C) ⊗ R)∗ (known as “the central
charge”), and a map τ from Alc to the set of bounded t-structures on C, satisfying certain compatibilities.
e (where Se is the
In [4], Anno, Bezrukavnikov and Mirkovic construst an example with C = Db (CohBe (S))
∗
pre-image to the Slodowy slice at e under the Springer map), V = hR , and Σ is the set of affine co-root
hyperplanes. The central charge Z : h∗ → (K 0 (C) ⊗ R)∗ is defined to be the unique polynomial map such
e λ ∈ Λ+ , Z(λ)[F ] is the Euler characteristic of F ⊗ O(λ). The map τ from Alc
that given F ∈ CohBe (S),
e is constructed in Section 1.8.2 of [10]; in Proposition
to the set of bounded t-structures on Db (CohBe (S))
1 of [4], it is proven that the axioms for a real variation of stability conditions are satisfied.
In an ongoing project, I plan to construct another example using certain sub-quotients of the principal
block O0 of BGG category O for a semi-simple Lie algebra g. Let Lw be the simple object with highest
weight w · 0; and for d ∈ Z+ , let O0<d (resp. O0≤d ) be the Serre subcategories containing Lw if its
Gelfand-Kirillov dimension is less than (resp. less than or equal to) d. We will be working with the Serre
quotient category C = O0d = O0≤d /O0<d , and the hyperplane arrangement with V = h∗R and Σ being the
set of linear co-root hyperplanes. Given λ ∈ Λ+ , M ∈ O0d , the central charge map Z is defined by letting
Z(λ)[M ] to be leading coefficient in a certain filtration polynomial for the module T0→λ M . Recalling that
Alc is in bijection with W , the map τ can be defined using the action of the braid group BW on Db (O0d ).
We claim that this data satisfies the axioms for a real variation of stability conditions.
3. Quiver varieties and the B(∞) crystal
Let g be a symmetrizable Kac-Moody algebra, and Uq (g) the corresponding quantum group. The B(∞)
(and B(λ)) crystals are combinatorial gadgets which encode much of the representation theory of Uq (g)
(for instance, tensor product multiplicities). In [15], Kashiwara and Saito geometrically realize the B(∞)
crystal as irreducible components of Lusztig’s quiver varieties.
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VINOTH NANDAKUMAR
We briefly recall their construction. Let Γ be a directed graph with vertex set I, and k edges between
vertices i and j, if hα̌i , αj i = −k. The pre-projective algebra Λ is the quotient of the path algebra CΓ by
the ideal generated by certain pre-projective relations for each vertex. Let Λ(ν) consist of representations
P
of Λ with weight vector ν ∈ Q+ = i∈I Z≥0 αi . One of the main results in [15] is that the set
G
Irr Λ(ν)
ν∈Q+
can be equipped with the structure of a crystal, and is isomorphic to B(∞).
3.1. Previous work. In [19] (joint with Peter Tingley), we extend this result to arbitrarily symmetrizable Lie algebras g. We consider a directed graph Γ with the additional datum of a finite field extension
Fi of Q for each vertex i ∈ I; and a (Fi , Fj ) bi-module for each edge τ between node i and node j. Given
a modulated graph, we can extract the Cartan datum and consider the corresponding symmetrizable
Kac-Moody algebra g. We can also define the path algebra CΓ by taking tensor products of bi-modules,
and define the pre-projective algebra Λ as the quotient of CΓ by certain canonical elements. Define Λ(ν)
to be representations of Λ with weight vector ν. Analogously to the above, in Theorem 4.7 of [19] we
show that
G
Irr Λ(ν)
ν∈Q+
can be equipped with the structure of a crystal, and is isomorphic to B(∞).
3.2. Future work. One caveat of the above construction is that it does not work over algebraically
closed fields, since we need to use field extensions of F; when we base change to its algebraic closure,
the resulting varieties no longer have the correct properties. It would be interesting to use this idea to
construct quiver varieties over C in non-symmetric type, by making some modifications to the varieties.
In [20], Saito gives a geometric construction of the crystal B(λ) for the highest weight irreducibles
of Uq (g) (where g is a symmetric Kac-Moody Lie algebra), by considering irreducible components of
Nakajima’s quiver varieties. It would be interesting to extend this result to arbitrary symmetrizable
Kac-Moody algebras, by using the techniques developed in [19]. This would be a first step towards giving
a construction of Nakajima’s quiver varieties in non-symmetric type.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, arXiv: math.RT/0601155.
S. Kato, An exotic Springer correspondence for symplectic groups, arXiv:math.RT/0607478.
R. Anno, Affine tangles and irreducible exotic sheaves, arXiv:0802.1070
R. Anno, R. Bezrukavnikov, I. Mirkovic, A thin stringy moduli space for Slodowy slices, arXiv:1108.1563.1
R. Anno, V. Nandakumar, Exotic t-structures for two-block Springer fibers
R. Bezrukavnikov, Cohomology of tilting modules over quantum groups and t-structures on derived categories of
coherent sheaves , Invent. Math. 166 (2006), 327–357
R. Bezrukavnikov, Perverse coherent sheaves, after Deligne, math.AG/0005152.
R. Bezrukavnikov, Perverse sheaves on affine flags and nilpotent cone of the Langlands dual group, Israel J. Math. 170
(2009), 185–206.
R. Bezrukavnikov, Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone, Represent. Theory
7 (2003)
R. Bezrukavnikov, I. Mirkovic, Representations of semi-simple Lie algebras in prime characteristic and non-commutative
Springer resolution
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[11] R. Bezrukavnikov, I. Mirkovic, D. Rumynin, Localization of modules for a semi-simple Lie algebra in prime characteristic
[12] R. Bezrukavnikov, S. Riche, Affine braid group actions on derived categories of Springer resolutions, arxiv:1101.3702
[13] S. Cautis, J. Kamnitzer, Knot homology via derived categories of coherent sheaves. I. The sl2 -case, Duke Math. J. 142
(2008), no. 3, 511–588.
[14] S. Cautis, J. Kamnitzer, A. Licata, Coherent sheaves on quiver varieties and categorification, Math. Ann. (2013), no.
3, 805-854
[15] M. Kashiwara, Y. Saito, Geometric constructions of crystal bases, Duke Mathematical Journal 89 (1997), no. 1, 9–36.
[16] A. Maffei, Quiver varieties of type A, arXiv:math/9812142
[17] H. Nakajima, Quiver varieties and finite-dimensional representations of quantum affine algebras. J. Amer. Math. Soc.
14 (2001), no.1, 145238.
[18] V. Nandakumar, Equivariant coherent sheaves on the exotic nilpotent cone, Represent. Theory 17 (2013), 663-681
[19] V. Nandakumar, P. Tingley, Lusztig’s quiver varieties and the B(∞) crystal in non-symmetric type
[20] Y. Saito, Crystal bases and quiver varieties, Math. Ann. 324, no. 4 (2002), 675-688
[21] N. Spaltenstein, The fixed point set of a unipotent transformation on the flag manifold, Indag. Math. 38 (1976), no.
5, 452-456
[22] R. Steinberg, On the desingularization of the unipotent variety, Invent. Math. 36 (1976), 209224.
[23] R. Steinberg, An occurrence of the RobinsonSchensted correspondence, J. Algebra 113 (1988), 523528.