Closed Form Multiplicity Polynomials Attached to
Hook-type Springer Fibers for SL(n, C).
Matthew Housley
housley@math.utah.edu
University of Utah
January 6th, 2011
Joint Mathematics Meetings - Special Session on Analytic and Geometric
Methods in Representation Theory
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Slides are available at www.math.utah.edu/~housley → talks.
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Setting
GR = SU(p, q)
gR = su(p, q)
g = sl(n, C), n = p + q
h = diagonal elements in g.
U(g) = universal enveloping algebra of g
Maximal compact:
KR = S(U(p) × U(q)), K = S(GL(p) × GL(q))
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Objects of study: irreducible (U(g), K )-modules.
(Assume regular integral infinitesimal character.)
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Commutative Algebra
I
Construct gr U(g) by using degree filtration.
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Commutative Algebra
I
I
Construct gr U(g) by using degree filtration.
PBW: gr U(g) ∼
= S(g).
(Algebra of polynomials in elements of g.)
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Commutative Algebra
I
I
Construct gr U(g) by using degree filtration.
PBW: gr U(g) ∼
= S(g).
(Algebra of polynomials in elements of g.)
Let X be an irreducible (U(g), K )-module at regular integral
infinitesimal character. We would like to define an
(S(g), K )-module.
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I
Let X0 be a K -invariant generating subspace of X .
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I
Let X0 be a K -invariant generating subspace of X .
I
Xn = Un (g) · X0 .
(Degree filtration.)
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I
Let X0 be a K -invariant generating subspace of X .
I
Xn = Un (g) · X0 .
(Degree filtration.)
I
Xn is K -invariant.
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I
Let X0 be a K -invariant generating subspace of X .
I
Xn = Un (g) · X0 .
(Degree filtration.)
I
Xn is K -invariant.
I
gr X is an (S(g/k), K )-module.
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I
Let X0 be a K -invariant generating subspace of X .
I
Xn = Un (g) · X0 .
(Degree filtration.)
I
Xn is K -invariant.
I
gr X is an (S(g/k), K )-module.
S(g/k) identifies with polynomial functions on (g/k)∗ .
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The structure of gr X depends on the choice of X0 .
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The structure of gr X depends on the choice of X0 .
Definition
(Vogan) Associated Variety:
AV(X ) = {λ ∈ (g/k)∗ |p(λ) = 0 for all p ∈ Ann(gr X )}
where Ann(X ) is the annihilator in S(g/k) of gr X .
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Associated Variety
Theorem
(Vogan) AV(X ) does not depend on the choice of X0 .
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Associated Variety
Theorem
(Vogan) AV(X ) does not depend on the choice of X0 .
The components of AV(X ) consist of closures of nilpotent K -orbits
on (g/k)∗ :
[
AV(X ) =
Oi
i
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Associated Cycle
AV(X ) =
[
Oi
i
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Associated Cycle
AV(X ) =
[
Oi
i
Oi ←→ Pi
where Pi is a minimal prime ideal in S(g/k) containing Ann(gr X ).
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Associated Cycle
AV(X ) =
[
Oi
i
Oi ←→ Pi
where Pi is a minimal prime ideal in S(g/k) containing Ann(gr X ).
I
Associated cycle: components of AV(X ) with multiplicity.
X
AC(X ) =
mi Oi
i
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Associated Cycle
AV(X ) =
[
Oi
i
Oi ←→ Pi
where Pi is a minimal prime ideal in S(g/k) containing Ann(gr X ).
I
Associated cycle: components of AV(X ) with multiplicity.
X
AC(X ) =
mi Oi
i
I
Roughly speaking, mi measures number of copies of
S(g/k)/Pi in gr X .
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Like AV(X ), AC(X ) is independent of our initial choices. We
would like to the calculate multiplicities mi .
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Let X be an irreducible (U(g), K )-module for SU(p, q) with trivial
infinitesimal character in the block of a finite dimensional
representation. (p 6= q : there is only one block.)
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Coherent Continuation
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Coherent Continuation
I
Coherent family: At each dominant regular integral weight,
we get a module X (λ), with X (ρ) = X . X (λ) has
infinitesimal character λ.
I
At other weights: virtual modules. (Formal sums of
irreducible modules.)
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Multiplicity Polynomials
I
In our setting, AV(X (λ)) is irreducible, so we only need to
calculate one integer to get AC(X (λ)).
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Multiplicity Polynomials
I
In our setting, AV(X (λ)) is irreducible, so we only need to
calculate one integer to get AC(X (λ)).
I
Let pX (λ) equal the multiplicity for AC(X (λ)) when λ is
dominant and regular.
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Multiplicity Polynomials
I
In our setting, AV(X (λ)) is irreducible, so we only need to
calculate one integer to get AC(X (λ)).
I
Let pX (λ) equal the multiplicity for AC(X (λ)) when λ is
dominant and regular.
Theorem
(Vogan) pX (λ) extends to a polynomial on h∗ .
New goal: calculate pX (λ).
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Cells
I
X is contained in an equivalence class of modules with the
same infinitesimal character called a Harish-Chandra cell. Let
X be in the cell C.
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Cells
I
X is contained in an equivalence class of modules with the
same infinitesimal character called a Harish-Chandra cell. Let
X be in the cell C.
I
Each Y ∈ C is in a coherent family and has an attached
multiplicity polynomial.
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Cells
I
X is contained in an equivalence class of modules with the
same infinitesimal character called a Harish-Chandra cell. Let
X be in the cell C.
I
Each Y ∈ C is in a coherent family and has an attached
multiplicity polynomial.
Definition
M(C) = spanC {pY (λ)|Y ∈ C}
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Cells
I
X is contained in an equivalence class of modules with the
same infinitesimal character called a Harish-Chandra cell. Let
X be in the cell C.
I
Each Y ∈ C is in a coherent family and has an attached
multiplicity polynomial.
Definition
M(C) = spanC {pY (λ)|Y ∈ C}
Theorem
(Joseph) M(C) is an irreducible W -representation.
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Strategy
I
Find one X ∈ C such that pX (λ) is easy to calculate.
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Strategy
I
Find one X ∈ C such that pX (λ) is easy to calculate.
I
Find pY (λ) for other modules in the cell by using the W
action on M(C).
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Question: how do we understand M(C) as a W -representation?
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Question: how do we understand M(C) as a W -representation?
Answer: we have the following isomorphisms:
Sp(C) ←− Coh(C) −→ M(C)
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Question: how do we understand M(C) as a W -representation?
Answer: we have the following isomorphisms:
Sp(C) ←− Coh(C) −→ M(C)
I
Sp(C) is a Springer representation: basis consists of
components of a certain Springer fiber.
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Question: how do we understand M(C) as a W -representation?
Answer: we have the following isomorphisms:
Sp(C) ←− Coh(C) −→ M(C)
I
Sp(C) is a Springer representation: basis consists of
components of a certain Springer fiber.
I
Coh(C) is a cell representation: basis consists of modules in C.
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Question: how do we understand M(C) as a W -representation?
Answer: we have the following isomorphisms:
Sp(C) ←− Coh(C) −→ M(C)
I
Sp(C) is a Springer representation: basis consists of
components of a certain Springer fiber.
I
Coh(C) is a cell representation: basis consists of modules in C.
I
These isomorphisms respect basis elements: each Y ∈ C maps
right to pY (λ) and left to a component Sp(Y ) of the Springer
fiber.
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Hook-type Case
I
Sp(C) is an irreducible representation of Sn . Such
representations are parameterized by Young diagrams with n
boxes.
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Hook-type Case
I
Sp(C) is an irreducible representation of Sn . Such
representations are parameterized by Young diagrams with n
boxes.
I
Let X ∈ C. We say that X is of hook type if Sp(C)
corresponds to a hook-type Young diagram. In this case,
Sp(C) is particularly easy to understand.
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Result
We get a formula for pX (λ) where X is a Harish-Chandra module
of hook-type:
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Result
We get a formula for pX (λ) where X is a Harish-Chandra module
of hook-type:

!
Y
X
X
1

sgn(σ 0 )σ 0 ·
zτm−i+1
pX (λ) = Am Q
sgn(σ)σ·
(i)
|τk |!
0
σ∈Sτ
σ ∈Sm
i=1...m
(zj are standard basis elements of the Cartan h.) The relevant
combinatorial parameters are obtained by looking at the Springer
fiber component Sp(X ).
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Details
Details are available at www.math.utah.edu/~housley →
research.
housley@math.utah.edu
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