MULTIPLICITY POLYNOMIALS ATTACHED TO HOOK TYPE
SPRINGER FIBERS
MATTHEW HOUSLEY
Abstract. We discuss a class of polynomials attached to Springer fiber components that has application to the computation of associated cycles for HarishChandra modules. The span of polynomials attached to components of a
specified fiber gives a Weyl group representation. In type A, this is isomorphic
to the Springer representation attached to the fiber and to a representation
on a cell of Harish-Chandra modules. We exploit these isomorphisms to write
down closed forms of polynomials for all hook type Springer fibers over sl(n, C).
These results build on work of Leticia Barchini and Roger Zierau [1].
1. Multiplicity Polynomials
A problem of current interest in geometric representation theory is the computation of associated cycles of Harish-Chandra modules [1]. Essentially, for a
Harish-Chandra module X, the associated cycle adds more detailed structure to
the associated variety. Let GR be a real Lie group and g = k ⊕ p the complexified
Cartan decomposition of its Lie algebra, with Nθ the cone of nilpotent elements
in p. The associated variety of X consists of a union of K-orbit closures in Nθ .
The associated cycle refines this by attaching an integer to each component of the
associated variety. As an invariant, the associated cycle often distinguishes HarishChandra modules that have the same associated varieties.
We restrict to the setting of GR = SU (p, q) and follow the exposition in the
appendix of [1]. We’ll let all our choices regarding GR be standard: the real group
is naturally defined as a set of linear transformations on Cp ⊕ Cq , so if g = k ⊕ p is
the complexified Lie algebra of GR , we let k be the set of maps preserving Cp and
Cq with p those maps that send Cp to Cq and vice versa; we choose the Cartan
subalgebra in g to be the set of diagonal matrices; and we determine a system of
positive roots by choosing a standard Borel subalgebra b to consist of all upper
triangular matrices in g.
Let X be a Harish-Chandra module for GR . The infinitesimal character of X
corresponds to some element λ in the dominant chamber of h∗ . We assume that λ
is dominant, integral and regular. The integrality condition means that λ is equal
to an integer linear combination of elements ei , where the ei are functions in h∗
that return the value of the ith diagonal entry of a matrix in h. From X, we can
build a coherent family of modules Φ. For each dominant integral regular weight
λ, Φ(λ) will be a Harish-Chandra module with infinitesimal character λ.
The associated variety of X is irreducible and each member of Φ corresponding
to a dominant regular integral weight has this same associated variety. Hence, the
associated cycle for each of these modules is the associated variety of X with an
integer attached. Define pX (λ) to be this integer for Φ(λ). The resulting function
1
2
MATTHEW HOUSLEY
is given by a harmonic polynomial on h∗ . Composing pX with the action of the
Weyl group W on h generates a Weyl group representation.
The module X is in a cell of Harish-Chandra modules as described in the appendix to [1]. The modules in this cell form a basis for a subquotient of the coherent
continuation representation.
Theorem 1.1. Let C(X) be the cell of Harish-Chandra modules containing X. The
map induced by
Y 7→ pY
for each Y ∈ C(X) is a W -equivariant isomorphism.
As mentioned above in the abstract, our strategy is to link this representation to
a Springer representation. Then, using the structures of the Springer representation
and the cell representation, we will derive a detailed description of the representation spanC {pY |Y ∈ C(X)}. Again following [1], we will find a particularly nice
Y ∈ C(X) for which pY (λ) is readily computable. Polynomials for other Y ∈ C(X)
will then be derived by the W -action on the pY .
To connect the W -representations we have so far to a Springer representation,
we define a set of polynomials qZ (λ) on h∗ . I’ve chosen my notation to follow
Joseph [5]. In our case, Z is an irreducible component of a Springer fiber. In
what follows, we will treat Z as a set of cosets in G/B, where B is the standard
Borel subgroup defined above. In addition, let vλ be a highest weight vector for an
irreducible finite dimensional representation of highest weight λ. Consider
dim spanC {gB · vλ |gB ∈ Z}.
As a function of vλ , this is asymptotic to a polynomial. (See section 1.9 of [5].)
We’ll call this this polynomial qZ (λ). In addition, we can take the fundamental
class attached to Z in the cohomology ring of G/B to the highest degree part gr qZ
of qZ . Extending this map to all components of a fixed Springer fiber gives a W equivariant isomorphism from the Springer representation to a representation on
the gr qZ , where this action is derived from the standard W -action on h∗ . Joseph
further proves in section 6.7 that
Z
gr qZ =
eλ .
Z
I do not attempt here to carefully define this integral, but see Joseph’s paper or [6]
near equation 5.6.
In the appendix to [1], Trapa shows how to attach a Springer fiber component
to each Harish-Chandra module for SU (p, q) with regular integral infinitesimal
character. To do this, we must look at the characteristic variety of the localization
of X. This variety consists of a union of closures of conormal bundles to K-orbits
on the flag variety B = G/B:
TQ∗ 1 B ∪ · · · ∪ TQ∗ k B.
The moment map image of this variety is the associated variety of X. We take
the leading term part of the characteristic variety over the single component of the
associated variety [9], i.e. the components TQ∗ i B that surject onto the associated
variety under the moment map. There is only one such component in this case. The
associated variety is the closure of a single nilpotent K-orbit. In this K-orbit, fix
an element f and take the Springer fiber over it. Define C(X) to be the component
MULTIPLICITY POLYNOMIALS ATTACHED TO HOOK TYPE SPRINGER FIBERS
3
of this Springer fiber contained in the leading term component of the characteristic
variety. Then,
Z
pX (λ) =
eλ .
C(X)
As a consequence,
pX (λ) = gr qC(X) (λ).
1.1. Summary of results from section 1. The Harish-Chandra module X has
irreducible associated variety containing a dense K-orbit. We pick an element f of
this dense K-orbit and construct a Springer fiber. The W -representation on the
cell C(X) of Harish-Chandra modules containing X is isomorphic to the Springer
representation attached to the Springer fiber over f and to a representation on polynomials gr qZ (λ) attached to components of this Springer fiber. These isomorphisms
respect basis elements in the sense that X in C(X) maps to the fundamental class of
the component C(X) in the Springer representation and gr qC(X) in the polynomial
representation. The associated cycle multiplicity for X will be given by gr qC(X) (λ),
with λ the infinitesimal character of X.
2. Structure of the Springer Representations attached to hook type
nilpotent orbits for type A groups
Let g be the lie algebra sl(n + 1, C). Fix a nilpotent element f ∈ g of hook type.
The moment map preimage of f is the Springer fiber F f . The fundamental classes
of the components of F f give a basis for the Springer representation parameterized
by the nilpotent orbit containing f .
These components are parameterized by standard Young tableaux with the
Young shape of f [7]. Taking B to be the set of all full flags in Cn+1 , F f consists
of flags preserved by the action of f . Given a standard tableau Y T with the Young
shape of f , consider the Young diagram Y Tk obtained by taking the boxes labelled
1 through k in Y T . We can look at flags preserved by f where the the restriction of
f to the kth subspace for each k from 1 to n + 1 has Young shape Y Tk . These flags
form a dense open subset of a component of F f and every component is determined
in this way by exactly one standard tableau.
Let W be the Weyl group of g. Let V be a representation of W . We define
the τ -invariant of a vector v ∈ V to be the set of all simple roots α such that
sα (v) = −v, where sα is reflection through α. If the Young diagram of f has m + 1
rows then the components of F f are in bijective correspondence with all sets of m
simple roots by letting an integer k below the top row of the tableau correspond to
the simple root ek−1 − ek .
For a simple root α, let CXα be the root space of α, CX−α the root space of −α
and CHα the span of the coroot of α. Define sα = CXα ⊕ CHα ⊕ CX−α . Let Sα
be the subgroup of G with Lie algebra sα . Treating G/B as the variety of Borel
subalgebras of G, let s ∈ Sα act by sending gBg −1 to gsBs−1 g −1 . Suppose that
α = ek − ek+1 . For an explicit flag F = V1 ⊂ V2 ⊂ · · · ⊂ Vn+1 , Sα · F equals the
one dimensional set of flags obtained by fixing all Vi except Vk , which can be any
subspace such that Vk−1 ⊂ Vk ⊂ Vk+1 . In particular if k + 1 appears below the top
row of a standard hook shaped tableau, then the action of Sek −ek+1 preserves the
corresponding Springer fiber component.
4
MATTHEW HOUSLEY
We will now use the structure of the orbital variety Of ∩ n, where Of is the
G-orbit on g containing f and n is the sum of the positive root spaces in g. Following [8], there is a correspondence between components of the flag variety and
the components of Of ∩ n. To construct the correspondence, we define two maps:
Let η : G → B be the projection from G to the flag variety and let π : G → Of
be given by g 7→ Adg (f ). Then π ◦ η −1 takes components of F f to components of
ON ∩ n. We also have an equivariance property for π ◦ η −1 : Define an action of Sα
on ON by taking Adg (N ) to Adgs (N ) for s ∈ Sα . Then π ◦ η −1 is Sα equivariant if
we let the action on the flag variety be the one defined in the previous paragraph.
In particular, Sα sends a component of Of ∩ n to itself if it sends the corresponding
Springer fiber component to itself.
In [4], Joseph defines a polynomial pC on h attached to a component C of Of ∩ n.
The lemma on page 244 of [4] tells us that sα in the Weyl group W acts on pC by
−1 when Sα sends C to itself. Theorem 4.2 in [8] says that the W -action on the
pC for the components in Of ∩ n gives a representation isomorphic to the Springer
representation for f , where the isomorphism takes pC to the fundamental class of
the component of F f corresponding to C.
Combining results from the previous few paragraphs, we now know a great deal
about the τ -invariants of fundamental classes of hook-type Springer fibers under
the Springer representation. Let αk = ek − ek+1 . We will show below that if the
tableau for a hook-type fiber has m + 1 rows, then the τ -invariant for any fiber
component under the Springer representation contains at most m-simple roots. It
follows that αk is in the τ -invariant for a certain component if and only if k + 1 is
below the top row of the standard tableau for that component.
We wish to completely characterize the action of W on the components of F f .
To do so, we use the fact that if f is of hook type and V
has m + 1 rows then
m
the Springer representation attached to f is isomorphic to
V , where V is the
standard representation of W [3, 2]. To study this representation, we choose basis
vectors of the form αa1 ∧ αa2 ∧ · · · ∧ αam , where the ak are in increasing order. We
give a few examples of the W -action on these vectors:
(1)
sα2 · (α1 ∧ α2 ∧ α3 )
= −(e1 − e3 ) ∧ α2 ∧ (e2 − e4 ) = −(α1 + α2 ) ∧ α2 ∧ (α2 + α3 )
= −α1 ∧ α2 ∧ α3 .
(2)
sα2 · (α1 ∧ α3 )
= (e1 − e3 ) ∧ (e2 − e4 ) = (α1 + α2 ) ∧ (α2 + α3 )
= α1 ∧ α3 + α1 ∧ α2 + α2 ∧ α3 .
If τ = {αa1 , . . . , αam } where the ak are in increasing order, let vτ = αa1 ∧ · · · ∧
αam . Now let τ be a set of m simple roots not containing αi . If αi−1 is an element
of τ , then let s+
αi (τ ) be the set obtained from τ by replacing αi−1 by αi . If αi+1
is an element of τ , then let s−
αi (τ ) be the set obtained from τ by replacing αi+1
with αi . If for instance s+
equal 0. The examples above
α (τ ) is undefined, let vs+
α (τ )
outline the proof of the following lemma:
MULTIPLICITY POLYNOMIALS ATTACHED TO HOOK TYPE SPRINGER FIBERS
5
Lemma 2.1. If α ∈ τ then sα (vτ ) = −vτ . If α ∈
/ τ , then
sα · vτ = vτ + vs+
+ vs−
.
α (τ )
α (τ )
With a little more work, we get this lemma:
Vm
Lemma 2.2. Let τ be a set of m simple roots. The set of vectors in
V with
τ -invariant τ is hvτ i \ {0}.
M
Proof. Let α be a simple root. Let Wα =
hvτ i. This space is invariant under
{τ |α∈τ }
Vm
Vm
sα , so we can define the quotient action of sα on (
V )/Wα . All of (
V )/Wα
is a +1 eigenspace for sα , so Wα must be the
entire
−1
eigenspace
for
the
action
\
Vm
of sα on
V . Given τ with m elements,
Wα = hvτ i.
α∈τ
Let Cτ be the component of F f with τ -invariant τ in the Springer
representation
Vm
attached to N . We now know that the isomorphism from
V to this Springer
representation takes vτ to some nonzero multiple of Cτ .
Define GR = SU (p, q) with (p, q) = (k, k) or (k, k + 1) so that p + q = n + 1.
Nilpotent K-orbits on gl(n + 1, C) are parameterized by signed Young tableau of
signature (p, q). For any hook shape, there will be a K-orbit Og whose signed
tableau has the correct shape. By remarks at the end of [1], there will be a HarishChandra module X whose associated variety is the closure of Og . Then, by section
1 the cell representation for the cell containing X is isomorphic to the Springer
representation attached to f with the isomorphism taking the basis of HarishChandra modules to the basis of Springer fiber components. We can thus apply
theorem 4.14 from [10]. In the present context, we have the following:
Lemma 2.3. Let τ and τ 0 be sets of m simple roots such that α ∈ τ , β ∈ τ 0 and
α∈
/ τ 0, β ∈
/ τ . Then, the multiplicity of Cτ in sα · Cτ 0 equals the multiplicity of Cτ 0
in sβ · Cτ and their common value is 0 or 1.
Combining this with the structural information we computed early, we get this:
Vm
Theorem 2.4. The linear map from
V to the Springer representation defined
by taking vτ to the fundamental class of Cτ is an isomorphism of W -representations.
3. Polynomial Computations
We wish to compute the polynomials gr qCτ (λ). For notational simplicity, we
will now call these pτ . We will find one component of each fiber where this is
computable using the dimension definition of pτ from the Joseph paper and then
derive the other polynomials for the fiber by using the structure of the Springer
representation computed above.
Let γ consist of the first m simple roots. Then, with an appropriate choice of f
Cγ = (SL(m + 1, C) × In−m ) · B, where B is the borel subalgebra of SL(n + 1, C)
corresponding to b. Let λ ∈ h∗ be a dominant integral regular weight with vλ a
highest weight vector of weight λ from an irreducible finite dimensional representation of sl(n+1, C). Then pγ (λ) is given by the highest degree part of the polynomial
asymptotic to
dim(spanC {(SL(m + 1, C) × In−m ) · vλ }).
6
MATTHEW HOUSLEY
We can compute pγ (λ) by the Weyl dimension formula. Let λ = x1 e1 + x2 e2 + . . . +
xn+1 en+1 Now, pγ becomes
Y
1
(xi − xj ).
m! · (m − 1)! · · · 1
1≤i<j≤m+1
Define Am =
1
m!·(m−1)!···1 .
Lemma 3.1.
(3)
pγ = Am
X
m−1
sgn(σ)σ · xm
· · · x1m x0m+1
1 x2
σ∈Sm+1
where the action of Sm+1 is on x1 , . . . , xm+1 .
Proof. To show this, we first useY
a little algebraic manipulation to see that the term
m m−1
1
(xi − xj ). Now, we will exploit the fact that
x1 x2
· · · xm appears in
1≤i<j≤m+1
Cτ 7→ pτ (λ) is an isomorphism. Then, for any α ∈ γ, sα ·pγ = −pγ . (This symmetry
is also visible in the original construction of pγ .) Hence pγ must contain all the
terms in (3). We need to establish that (3) is not missing any terms. Notice that sα ,
α ∈ γ must not act as identity on any term. If more than one of x1 , . . . , xm+1 are
missing from a term, we can apply the sα to generate a term such that the missing
variables are x1 and x2 . The reflection sα1 acts as one on this term, a contradiction.
Suppose that two variables in a term have the same power. Once again, we can
apply the sα to make these two variables adjacent and yield a contradiction. By
considering the homogeneous degree of pγ , we see that each term must have the
m−1
· · · x1im with no two ik alike.
form xm
i1 xi2
Lemma 3.2. For any τ consisting of m simple roots, pτ consists of terms of the
m−1
form xm
· · · x1im with the ik distinct.
i1 xi2
Proof. The pτ are generated from pγ by permuting the variables x1 , x2 . . . xn+1 and
taking linear combinations.
m−1
· · · x1im (with b a constant) is a term of pτ , then so
Lemma 3.3. If b · xm
i1 x i2
1
m m−1
is sgn(σ)bσ · xi1 xi2 · · · xim , where σ is any permutation acting on the variables
xi1 , xi2 , . . . xim .
Proof. This is obvious for pγ . We again use the fact that the pτ are generated from
pγ by permuting variables and taking linear combinations.
Before we state and prove the general form for pτ , we will need a few more
definitions and pieces of notations. We begin by defining the components of τ .
The elements of τ define root subsystem of An . This subsystem defines a Dynkin
subdiagram of the Dynkin diagram for An . (This is given by the nodes in τ with
edges between adjacent nodes.) We define the components of τ to be sets of simple
roots corresponding to the connected components of the subdiagram for τ . If we
parameterize the components by an index i, let |τi | be the number of elements in
the i-th component.
Define τ (k) to be the index of the k-th simple root in τ from the left, i.e. if
τ = {α1 , α3 , α4 }, then τ (3) = 4. Define Sτ to be the subgroup of Sn+1 generated
by the simple reflections parameterized by the elements of τ .
MULTIPLICITY POLYNOMIALS ATTACHED TO HOOK TYPE SPRINGER FIBERS
7
Theorem 3.4.
(4)
X
1
sgn(σ)σ ·
pτ = Am Q
|τk |!
σ∈Sτ
!!
X
0
0
sgn(σ )σ ·
σ 0 ∈Sm
Y
xm−i+1
τ (i)
i=1...m
where Sm acts only on the m variables that appear in the inner
all the variables.
Q
and Sτ acts on
Proof. First, we showQthat this formula agrees with our earlier formula for τ = γ.
The action of Sm on i=1...m xm−i+1
generates m! distinct terms. Each of these is
τ (i)
Q
m−i+1
mapped back to i=1...m xτ (i)
by a unique permutation in Sγ . The coefficient
Q
1/|γ|! cancels this extra multiplicity. From now on, we will call i=1...m xm−i+1
τ (i)
the generating term of pτ .
We must show that the generating term in each pτ has coefficient Am . We
extend the Q
action of Sm to all the variables by letting it act by identity
on variables
Q
1
not inside . Then, the intersection of Sm with Sτ has order |τk |! and the Q
coefficient cancels this.
We proceed by induction. Our induction hypothesis is that the theorem is true
for any τ 0 that can be constructed from τ by shifting simple roots to the left on the
0
Dynkin diagram. We may assume that there exist αj and τ 0 such that s+
αj (τ ) = τ .
0
We’ll work out two cases. First, assume that s−
αj (τ ) does not exist. In this case,
0
we know that τ does not contain αj or αj+1 but contains αj−1 ; that τ contains αj
but not αj−1 or αj+1 ; and that any other root in τ is in τ 0 and vice versa. Then,
sαj · pτ 0 = pτ 0 + ps+
α
j
(τ 0 )
= pτ 0 + pτ .
In other words,
pτ = sαj · pτ 0 − pτ 0 .
Working carefully with formula (4), we see that pτ 0 does not contain the variable
xj+1 . Thus, sαj · pτ 0 does not contain the variable xj and cannot contain a term of
Q
the form i=1...m xm−i+1
. On the other hand, starting with the generating term
τ (i)
for pτ 0 and acting by −sαj−1 yields −1 times the generating term for pτ , so pτ must
Q
contain a term of the form Am i=1...m xm−i+1
.
τ (i)
0
−
Now, we move to the case when sαj (τ ) is defined. We have the equation
pτ = sαj · pτ 0 − pτ 0 − ps−
α
j
(τ 0 ) .
0
In this case, τ 0 contains αj−1 and αj+1 ; s−
αj τ contains αj−1 and αj but not αj+1 ;
τ contains αj+1 and αj but not αj−1 ; and any other root appearing in any one of
these τ -invariants appears in all three.
Applying formula (4) carefully, we see that sαj · pτ 0 contains a term of the form
Q
Q
Am i=1...m xm−i+1
; pτ 0 contains a term of the form −Am i=1...m xm−i+1
; and
τ (i)
τ (i)
Q
m−i+1
ps−
contains a term of the form Am i=1...m xτ (i) . Thus, coefficients for
0
αj (τ )
Q
terms of type i=1...m xm−i+1
match on the two sides of the equation.
τ (i)
Now that we know we have the correct generating term, we apply the τ -invariant
and lemma 3.3 which give us the Sm and Sτ actions in the formula. Thus, all the
terms in the formula must appear in pτ . We must show that no terms are missing.
First, observe that the basic term form is given by lemma 3.2. Thus, each term
8
MATTHEW HOUSLEY
must contain m distinct variables. Now consider the component structure of τ . If
the component τk contains simple roots αj , . . . , αj 0 , then this component acts on
the variables xj , . . . , xj 0 , xj 0 +1 . As discussed in the proof of lemma 3.1, if two or
more variables in this set do not appear in a term, this gives rise to a term such
that sα for some α in τk acts by identity. Thus, each term is missing at most one of
the variables xj , . . . , xj 0 , xj 0 +1 . Using the fact that each term contains m variables,
each term is missing at least one of the variables xj , . . . , xj 0 , xj 0 +1 . Thus, each term
is missing exactly one of xj , . . . , xj 0 , xj 0 +1 . Combining this with 3.2, we see that
equation (4) generates terms of all possible forms.
4. A Sample Calculation
Here, we will reproduce the calculation in example 6.10 of [1] using the closed
form above. Let p = 3 so that our real form is SU (3, 2). Let Z be the closed
K-orbit on B discussed in the example. In [1], Z is parameterized by an array of
dots with attached integers. We can also attach to Z a sequence of plus and minus
signs as described in [11]. In this parameterization, Z is attached to the sequence
(− + + + −). A third parameterization exists wherein Z is attached to a pair of
tableaux, one signed and one standard[8]. In this picture, the tableaux are
−+−
+
+
1 2 5
3
4
.
The moment map image of OZ of Z is the nilpotent K-orbit in p parameterized
by the signed tableau in the pair above. If we fix an element f ∈ OZ , the Springer
fiber F f intersected with Z is the component of F f parameterized by the standard
tableau in the pair above. Thus, the τ -invariant for the component is {e2 − e3 , e3 −
e4 }. Then, using equation (4), we have
!!
X
X
Y
1
m−i+1
0
0
pτ = Am Q
sgn(σ)σ ·
sgn(σ )σ ·
xτ (i)
|τk |!
0
i=1...m
σ ∈Sm
σ∈Sτ
11 X
=
sgn(σ)σ ·
22
σ∈Sτ
=
!
X
0
0
sgn(σ )σ ·
(x22 x3 )
σ 0 ∈Sm
1 X
sgn(σ)σ · (x22 x3 − x23 x2 )
4
σ∈Sτ
(5)
=
1 2
(x x3 − x23 x2 + x24 x2 − x22 x4 + x23 x4 − x24 x3 ).
2 2
Now we’ll follow the polynomial computation in [1]. Note that in the closed form
computation above, we are using the standard positive system for gl(5, C). This
will be important in a moment. In example 6.10, note that we are not using
the standard positive system. Instead, we’re using the system with simple roots
{e4 − e1 , e1 − e2 , e2 − e3 , e3 − e5 }. The polynomial is given by dim(L · w(λ+ρ−2ρc ) )
where wλ is a highest weight for irreducible representation of SL(5, C) with highest
weight λ. In this case, L = GL(3, C) × GL(1, C) × GL(1, C). The first copy of
GL(3, C) corresponds to the Lie algebra generated by the simple roots e1 − e2 and
MULTIPLICITY POLYNOMIALS ATTACHED TO HOOK TYPE SPRINGER FIBERS
9
e2 − e3 . We apply the Weyl dimension formula to obtain the expression
1 Y
p(λ) =
(λ + 2ρ − 2ρc , ei − ej ).
2
1≤i<j≤3
ρ − ρc = e1 + e2 + e3 − 2e4 − e5 . As a result, p(λ) simplifies to
1 Y
1
(6)
p(λ) =
(λ, ei − ej ) = (x1 − x2 )(x1 − x3 )(x2 − x3 ).
2
2
1≤i<j≤3
At first glance, this expresssion is problematic because it does not match (5). The
problem is that we made different choices of positive system in the two multiplicity
computations. To fix this, we need to construct the Weyl group element that takes
dominant weights for the standard positive system to dominant weights for the
simple roots {e4 − e1 , e1 − e2 , e2 − e3 , e3 − e5 }. The correct Weyl group element in
cycle notation is (1432). Letting this act on h∗ and composing with p(λ), we obtain
1
1
(x2 −x3 )(x2 −x4 )(x3 −x4 ) = (x22 x3 −x23 x2 +x24 x2 −x22 x4 +x23 x4 −x24 x3 ) = pτ (λ).
2
2
References
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discrete series representations of SU(p, q). Represent. Theory, 12:403–434, 2008. With an
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[2] David H. Collingwood and William M. McGovern. Nilpotent orbits in semisimple Lie algebras.
Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York, 1993.
[3] William Fulton and Joe Harris. Representation theory, volume 129 of Graduate Texts in
Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.
[4] Anthony Joseph. On the variety of a highest weight module. J. Algebra, 88(1):238–278, 1984.
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