functors for matching kazhdan-lusztig polynomials for gl( n;f
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Functors for MatchingKazhdan-Lusztig polynomials for GL(n; F )
Peter E. Trapa(joint with Dan Ciubotaru)
F� : X 7! H0(X F )[�+N�]
KLV polynomials
GR real reductive group, g = gR R CKR maximal compact subgroup, K = (KR)CB variety of Borel subalgebras in g.LocK(B) set of irreducible K-equivariant locally constantsheaves supported on single K orbits on B
Given L;L0 2 LocK(B), we can de�ne theKazhdan-Lusztig-Vogan polynomial
pLL0(q):
KLV polynomials
GR real reductive group, g = gR R CKR maximal compact subgroup, K = (KR)CB variety of Borel subalgebras in g.LocK(B) set of irreducible K-equivariant locally constantsheaves supported on single K orbits on B
Given L;L0 2 LocK(B), we can de�ne theKazhdan-Lusztig-Vogan polynomial
pLL0(q):
KLV polynomials
GR real reductive group, g = gR R CKR maximal compact subgroup, K = (KR)CB variety of Borel subalgebras in g.LocK(B) set of irreducible K-equivariant locally constantsheaves supported on single K orbits on B
Given L;L0 2 LocK(B), we can de�ne theKazhdan-Lusztig-Vogan polynomial
pLL0(q):
Notation for special case
If L and L0 are the constant sheaves on Q;Q0 2 KnB, insteadwrite
pQ;Q0 = pL;L0
Notation for an even more special case
If GR is already a complex Lie group, then the Bruhatdecomposition identi�es KnB with the Weyl group W of g.
Infact,
LocK(B) $ W:
If Q(x) and Q(y) are two orbits parametrized by x and y, thenwe write
px;y = pQ(x);Q(y):
Notation for an even more special case
If GR is already a complex Lie group, then the Bruhatdecomposition identi�es KnB with the Weyl group W of g. Infact,
LocK(B) $ W:
If Q(x) and Q(y) are two orbits parametrized by x and y, thenwe write
px;y = pQ(x);Q(y):
Notation for an even more special case
If GR is already a complex Lie group, then the Bruhatdecomposition identi�es KnB with the Weyl group W of g. Infact,
LocK(B) $ W:
If Q(x) and Q(y) are two orbits parametrized by x and y, thenwe write
px;y = pQ(x);Q(y):
Another example
GR = U(p; q)
KR = U(p)�U(q)K = GL(p;C)�GL(q;C)
In factLocK(B) $ (GL(p;C)�GL(q;C))nB:
Another example
GR = U(p; q)KR = U(p)�U(q)
K = GL(p;C)�GL(q;C)
In factLocK(B) $ (GL(p;C)�GL(q;C))nB:
Another example
GR = U(p; q)KR = U(p)�U(q)K = GL(p;C)�GL(q;C)
In factLocK(B) $ (GL(p;C)�GL(q;C))nB:
Another example
GR = U(p; q)KR = U(p)�U(q)K = GL(p;C)�GL(q;C)
In factLocK(B) $ (GL(p;C)�GL(q;C))nB:
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.
Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
What does \matching KLV polynomials" mean?
Let G1R and G2
R denote two real groups.Suppose we can �nd
(1) a subset S1 � LocK1(B1);
(2) a subset S2 � LocK2(B2); and
(3) a bijection � : S1 ! S2
such that for any L;L0 2 S1,
pL;L0 = p�(L);�(L0):
Then we say � implements a matching of KLV polynomials.
Obviously this is more impressive if S1 is large.
Plan
� Explicitly describe a matching for GL(n;C) with variousU(p; q)'s, p+ q = n.
� The key will be an intermediate combinatorial set (which isreally GL(n;Qp) in disguise).
� Indicate brie y a general setting to search for suchmatchings. (Works in all simple adjoint classical cases, forinstance, but breaks in interesting ways in a handful ofexceptional ones.)
� Hando� to Dan: a functor \implementing" the matching.
Plan
� Explicitly describe a matching for GL(n;C) with variousU(p; q)'s, p+ q = n.
� The key will be an intermediate combinatorial set (which isreally GL(n;Qp) in disguise).
� Indicate brie y a general setting to search for suchmatchings. (Works in all simple adjoint classical cases, forinstance, but breaks in interesting ways in a handful ofexceptional ones.)
� Hando� to Dan: a functor \implementing" the matching.
Plan
� Explicitly describe a matching for GL(n;C) with variousU(p; q)'s, p+ q = n.
� The key will be an intermediate combinatorial set (which isreally GL(n;Qp) in disguise).
� Indicate brie y a general setting to search for suchmatchings. (Works in all simple adjoint classical cases, forinstance, but breaks in interesting ways in a handful ofexceptional ones.)
� Hando� to Dan: a functor \implementing" the matching.
Plan
� Explicitly describe a matching for GL(n;C) with variousU(p; q)'s, p+ q = n.
� The key will be an intermediate combinatorial set (which isreally GL(n;Qp) in disguise).
� Indicate brie y a general setting to search for suchmatchings. (Works in all simple adjoint classical cases, forinstance, but breaks in interesting ways in a handful ofexceptional ones.)
� Hando� to Dan: a functor \implementing" the matching.
Plan
� Explicitly describe a matching for GL(n;C) with variousU(p; q)'s, p+ q = n.
� The key will be an intermediate combinatorial set (which isreally GL(n;Qp) in disguise).
� Indicate brie y a general setting to search for suchmatchings. (Works in all simple adjoint classical cases, forinstance, but breaks in interesting ways in a handful ofexceptional ones.)
� Hando� to Dan: a functor \implementing" the matching.
Intermediate Combinatorial Set
Fix � = (�01 � � � � � �0n) (an in�nitesimal character in disguise).
Everything reduces to the case
1 �0i 2 Z
2 �0i � �0i+1 2 f0; 1g.
We assume this is the case and rewrite
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Intermediate Combinatorial Set
Fix � = (�01 � � � � � �0n) (an in�nitesimal character in disguise).
Everything reduces to the case
1 �0i 2 Z
2 �0i � �0i+1 2 f0; 1g.
We assume this is the case and rewrite
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Intermediate Combinatorial Set
Fix � = (�01 � � � � � �0n) (an in�nitesimal character in disguise).
Everything reduces to the case
1 �0i 2 Z
2 �0i � �0i+1 2 f0; 1g.
We assume this is the case and rewrite
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Intermediate Combinatorial Set
Fix � = (�01 � � � � � �0n) (an in�nitesimal character in disguise).
Everything reduces to the case
1 �0i 2 Z
2 �0i � �0i+1 2 f0; 1g.
We assume this is the case and rewrite
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Let W (�) = stabilizer of � under the permutation action ofW = Sn. So
W (�) = Sn1 � � � � � � � � � � � Snk :
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Let W (�) = stabilizer of � under the permutation action ofW = Sn.
SoW (�) = Sn1 � � � � � � � � � � � Snk :
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Let W (�) = stabilizer of � under the permutation action ofW = Sn. So
W (�) = Sn1 � � � � � � � � � � � Snk :
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Consider a collection of k piles of dots, with the ith pileconsisting of ni dots labeled �i.For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), consider:
5 4 3 2 1�
��
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Consider a collection of k piles of dots, with the ith pileconsisting of ni dots labeled �i.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), consider:
5 4 3 2 1�
��
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
� = (
n1z }| {�1; : : : ; �1 > � � � � � � � � � >
nkz }| {�k; : : : ; �k):
Consider a collection of k piles of dots, with the ith pileconsisting of ni dots labeled �i.For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), consider:
5 4 3 2 1�
��
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
��
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
��
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
��
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
��
��
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
��
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we cannotconsider the following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
� ��
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we cannotconsider the following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we cannotconsider the following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
jjjjjjjjjjjjjjjjj
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graph:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
Intermediate Combinatorial Set
Consider a directed graph such that on this vertex set such that
� The only allowable edges are of the form xi ! xi+1 with xiin the �i pile and xi+1 in the �i+1 pile; and
� No two edges originate at the same vertex.
For instance, if � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1), we can considerthe following graphs:
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
Clearly W (�) ' Sn1 � � � � � Snk acts on such graphs.
Intermediate Combinatorial Set
For instance,
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
and 5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
are in the same S2 � S4 � S3 � S2 � S1 orbit.
The set of equivalence classes, denoted MS(�), arecalled �-multisegments.
Intermediate Combinatorial Set
For instance,
5 4 3 2 1�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo�
�
OOOOOO�
�ooo
ooo
�
�
�
�
and 5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
are in the same S2 � S4 � S3 � S2 � S1 orbit.
The set of equivalence classes, denoted MS(�), arecalled �-multisegments.
How do multisegments naturally arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h�, and consider
1 G(�) = Ad-centralizer of � in G.
2 g1(�) = +1-eigenspace of ad(�) on g
Except this doesn't make sense.
How do multisegments naturally arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h�, and consider
1 G(�) = Ad-centralizer of � in G.
2 g1(�) = +1-eigenspace of ad(�) on g
Except this doesn't make sense.
How do multisegments naturally arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h�, and consider
1 G(�) = Ad-centralizer of � in G.
2 g1(�) = +1-eigenspace of ad(�) on g
Except this doesn't make sense.
How do multisegments naturally arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h�, and consider
1 G(�) = Ad-centralizer of � in G.
2 g1(�) = +1-eigenspace of ad(�) on g
Except this doesn't make sense.
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Try again: How do multisegments naturally
arise?
Let G = GL(n;C), g = gl(n;C), view � 2 h� ' h_, and consider
1 G_(�) = Ad-centralizer of � in G_.
2 g_1 (�) = +1-eigenspace of ad_(�) on g_
Then G_(�) naturally acts on g_1 (�).
MS(�) parametrizes these orbits.
There is a beautiful generalization to all types (Kawanaka,Lusztig, Vinberg).
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1).
So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1b1 c1c1 d1d1 e1
b2 c2c2 d2a2
b3b4
c3
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1). So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1b1 c1c1 d1d1 e1
b2 c2c2 d2a2
b3b4
c3
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1). So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1b1 c1c1 d1d1 e1
b2 c2c2 d2a2
b3b4
c3
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1). So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1b1 c1c1 d1d1 e1
b2 c2c2 d2a2
b3b4
c3
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1). So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1b1 c1c1 d1d1 e1
b2 c2c2 d2a2
b3b4
c3
Example of parametrization
Take � = (5; 5; 4; 4; 4; 4; 3; 3; 3; 2; 2; 1). So
G_(�)'GL(2;C)�GL(4;C)�GL(3;C)�GL(2;C)�GL(1;C)
g_1 (�) 'nC2 A
�! C4 B�! C3 C
�! C2 D�! C1
o
For instance,
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1a1 b1
� //b1 c1� //c1 d1
� //d1 e1� //
b2 c2� //c2 d2
� //a2
b3b4
c3
Another kind of KL polynomial
Since MS(�) parametrizes orbits of G_(�) on g_1 (�), givens; s0 2MS(�) we can consider
ps;s0(q) := pL;L0(q)
where L;L0 2 LocG_(�)(g_1 (�)) are the constant sheaves on the
orbits parametrized by s; s0.
In general, Lusztig (2006) has given an algorithm to computethese polynomials.
Another kind of KL polynomial
Since MS(�) parametrizes orbits of G_(�) on g_1 (�), givens; s0 2MS(�) we can consider
ps;s0(q) := pL;L0(q)
where L;L0 2 LocG_(�)(g_1 (�)) are the constant sheaves on the
orbits parametrized by s; s0.
In general, Lusztig (2006) has given an algorithm to computethese polynomials.
Another kind of KL polynomial
Since MS(�) parametrizes orbits of G_(�) on g_1 (�), givens; s0 2MS(�) we can consider
ps;s0(q) := pL;L0(q)
where L;L0 2 LocG_(�)(g_1 (�)) are the constant sheaves on the
orbits parametrized by s; s0.
In general, Lusztig (2006) has given an algorithm to computethese polynomials.
The plan
We are going to de�ne injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn
such that all Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials forGL(n;C) and U(p; q).
The plan
We are going to de�ne injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn
such that all Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials forGL(n;C) and U(p; q).
The plan
We are going to de�ne injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn
such that all Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials forGL(n;C) and U(p; q).
The plan
We are going to de�ne injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn
such that all Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials forGL(n;C) and U(p; q).
The plan
We are going to de�ne injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn
such that all Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials forGL(n;C) and U(p; q).
Defining �1 : MS(�) �! Sn
� �� �� �� �
� �� ��
�
�
�
1 33 77 1010 12
4 88 112
5
6
9
Set � = (1 3 4 7 10 12)(4 8 11): And
�1(s) := longest element in �W (�):
Defining �1 : MS(�) �! Sn
� �� �� �� �
� �� ��
�
�
�
1 33 77 1010 12
4 88 112
5
6
9
Set � = (1 3 4 7 10 12)(4 8 11): And
�1(s) := longest element in �W (�):
Defining �1 : MS(�) �! Sn
� �� �� �� �
� �� ��
�
�
�
1 33 77 1010 12
4 88 112
5
6
9
Set � = (1 3 4 7 10 12)(4 8 11): And
�1(s) := longest element in �W (�):
Defining �1 : MS(�) �! Sn
� �� �� �� �
� �� ��
�
�
�
1 33 77 1010 12
4 88 112
5
6
9
Set � = (1 3 4 7 10 12)(4 8 11):
And
�1(s) := longest element in �W (�):
Defining �1 : MS(�) �! Sn
� �� �� �� �
� �� ��
�
�
�
1 33 77 1010 12
4 88 112
5
6
9
Set � = (1 3 4 7 10 12)(4 8 11): And
�1(s) := longest element in �W (�):
De�ning
�2 : MS(�) �!`
p+q=n(GL(p;C)�GL(q;C))nBn:
Let
��(n) = finvolutions in Sn with signed �xed pointsg:
Then there is a bijection
��(n)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
De�ning
�2 : MS(�) �!`
p+q=n(GL(p;C)�GL(q;C))nBn:
Let
��(n) = finvolutions in Sn with signed �xed pointsg:
Then there is a bijection
��(n)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
De�ning
�2 : MS(�) �!`
p+q=n(GL(p;C)�GL(q;C))nBn:
Let
��(n) = finvolutions in Sn with signed �xed pointsg:
Then there is a bijection
��(n)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms represented by
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X = f(z; T 0; B0) j T 0 � B0; z2 2 Z(G); xT 0x�1 = T 0g=G
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .
Fix the inner class of (classical) real forms represented by
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X = f(z; T 0; B0) j T 0 � B0; z2 2 Z(G); xT 0x�1 = T 0g=G
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms represented by
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X = f(z; T 0; B0) j T 0 � B0; z2 2 Z(G); xT 0x�1 = T 0g=G
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms represented by
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X = f(z; T 0; B0) j T 0 � B0; z2 2 Z(G); xT 0x�1 = T 0g=G
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X (e) = f(z; T 0; B0) j T 0 � B0; z2 = e; xT 0x�1 = T 0g=G
= fz 2 NG(T ) j z2 = eg=T $ ��(n):
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X (e) = f(z; T 0; B0) j T 0 � B0; z2 = e; xT 0x�1 = T 0g=G
= fz 2 NG(T ) j z2 = eg=T
$ ��(n):
A few details on the ��(n) parametrization.
Set G = GL(n;C) and �x a torus T .Fix the inner class of (classical) real forms
fU(p; q) j p+ q = n; p � qg:
Consider its one-sided parameter space
X (e) = f(z; T 0; B0) j T 0 � B0; z2 = e; xT 0x�1 = T 0g=G
= fz 2 NG(T ) j z2 = eg=T $ ��(n):
A few details on the ��(n) parametrization.
Recall the reason for introducing X (e):
X (e) $ai
KinBn
where the union is over all strong real forms (lying over e).
Our case (G conjugacy classes of elements of order 2):
X (e)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
A few details on the ��(n) parametrization.
Recall the reason for introducing X (e):
X (e) $ai
KinBn
where the union is over all strong real forms (lying over e).
Our case (G conjugacy classes of elements of order 2):
X (e)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
A few details on the ��(n) parametrization.
Recall the reason for introducing X (e):
X (e) $ai
KinBn
where the union is over all strong real forms (lying over e).
Our case (G conjugacy classes of elements of order 2):
X (e)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
A few details on the ��(n) parametrization.
Recall the reason for introducing X (e):
X (e) $ai
KinBn
where the union is over all strong real forms (lying over e).
Our case (G conjugacy classes of elements of order 2):
��(n)$a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points) = 6 +
4
2= 8:
q = (# of � signs) +1
2(# non-�xed points) = 2 +
4
2= 4:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points) = 6 +
4
2= 8:
q = (# of � signs) +1
2(# non-�xed points) = 2 +
4
2= 4:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points)
= 6 +4
2= 8:
q = (# of � signs) +1
2(# non-�xed points) = 2 +
4
2= 4:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points) = 6 +
4
2= 8:
q = (# of � signs) +1
2(# non-�xed points) = 2 +
4
2= 4:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points) = 6 +
4
2= 8:
q = (# of � signs) +1
2(# non-�xed points)
= 2 +4
2= 4:
Notation
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
Read o�
p = (# of + signs) +1
2(# non-�xed points) = 6 +
4
2= 8:
q = (# of � signs) +1
2(# non-�xed points) = 2 +
4
2= 4:
Notation
A convenient short-hand (as in Monty's talk): Rewrite
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
as+ 1 2 + + � � + + + 2 1
Notation
A convenient short-hand (as in Monty's talk):
Rewrite
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
as+ 1 2 + + � � + + + 2 1
Notation
A convenient short-hand (as in Monty's talk): Rewrite
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
as+ 1 2 + + � � + + + 2 1
Notation
A convenient short-hand (as in Monty's talk): Rewrite
+ � ++� ,,+ + � � + + + �rr �ss 2 ��(12):
as+ 1 2 + + � � + + + 2 1
A few comments on the ��(n) parametrization.
General features:
1 This parametrization is \dual" to the one forLocO(n;C)(Bn), i.e. the representation theory of GL(n;R).
2 The closed orbits are parametrized by diagrams consistingof all signs.
3 It's easy to translate the information from the atlascommand kgb in this parametrization.
A few comments on the ��(n) parametrization.
General features:
1 This parametrization is \dual" to the one forLocO(n;C)(Bn), i.e. the representation theory of GL(n;R).
2 The closed orbits are parametrized by diagrams consistingof all signs.
3 It's easy to translate the information from the atlascommand kgb in this parametrization.
A few comments on the ��(n) parametrization.
General features:
1 This parametrization is \dual" to the one forLocO(n;C)(Bn), i.e. the representation theory of GL(n;R).
2 The closed orbits are parametrized by diagrams consistingof all signs.
3 It's easy to translate the information from the atlascommand kgb in this parametrization.
A few comments on the ��(n) parametrization.
General features:
1 This parametrization is \dual" to the one forLocO(n;C)(Bn), i.e. the representation theory of GL(n;R).
2 The closed orbits are parametrized by diagrams consistingof all signs.
3 It's easy to translate the information from the atlascommand kgb in this parametrization.
A few comments on the ��(n) parametrization.
General features:
1 This parametrization is \dual" to the one forLocO(n;C)(Bn), i.e. the representation theory of GL(n;R).
2 The closed orbits are parametrized by diagrams consistingof all signs.
3 It's easy to translate the information from the atlascommand kgb in this parametrization.
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������+�+
+11
�
??????????????
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������+�+
+11
�
??????????????
11+
�++
11+
�
??????????????
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������+�+
+11
�
??????????????
11+
�++
11+
�
??????????????+�+
11+
�
��������������
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������+�+
+11
�
??????????????
11+
�++
11+
�
??????????????+�+
11+
�
��������������
1 + 1
+11
1 + 1
�
��������������
An example of the parametrization for
(p; q) = (2; 1).
Set � = e1 � e2, � = e2 � e3, and consider(GL(2;C)�GL(1;C))nB3.
+ +� +�+ �++
+11
+ +�
+11
�
��������������+�+
+11
�
??????????????
11+
�++
11+
�
??????????????+�+
11+
�
��������������
1 + 1
+11
1 + 1
�
��������������11+
1 + 1
�
??????????????
De�ning �2 : MS(�) �! ��(n)
Consider our running example:
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1+ �� ++ �� +
� ++ �+
�
�
+
De�ning �2 : MS(�) �! ��(n)
Consider our running example:
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1+ �� ++ �� +
� ++ �+
�
�
+
De�ning �2 : MS(�) �! ��(n)
Consider our running example:
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1� + � + �
� ++ �+
�
�
+
De�ning �2 : MS(�) �! ��(n)
Consider our running example:
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1� � � + �
� ++ �+
�
�
+
Now \ atten".
De�ning �2 : MS(�) �! ��(n)
Consider our running example:
5 4 3 2 1� �� �� �� �
� �� ��
�
�
�
5 4 3 2 1� � � + �
� ++ �+
�
�
+
Now \ atten".
Flatten...
� ))+ � � � � ((� + + �vv+ �uu
The attening procedure is not well-de�ned.
Remedy: take largest dimensional orbit obtained this way.
Flatten...
� ))+ � � � � ((� + + �vv+ �uu
The attening procedure is not well-de�ned.
Remedy: take largest dimensional orbit obtained this way.
Flatten...
� ))+ � � � � ((� + + �vv+ �uu
The attening procedure is not well-de�ned.
Remedy: take largest dimensional orbit obtained this way.
The upshot
We have de�ned injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Theorem (Zelevinsky, CT)
All Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials for
GL(n;C) and U(p; q).
The upshot
We have de�ned injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Theorem (Zelevinsky, CT)
All Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials for
GL(n;C) and U(p; q).
The upshot
We have de�ned injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Theorem (Zelevinsky, CT)
All Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials for
GL(n;C) and U(p; q).
The upshot
We have de�ned injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Theorem (Zelevinsky, CT)
All Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials for
GL(n;C) and U(p; q).
The upshot
We have de�ned injections
�1 : MS(�) �! Sn
and
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Theorem (Zelevinsky, CT)
All Kazhdan-Lusztig-Vogan polynomials match:
ps;s0 = p�1(s);�1(s0) = p�2(s);�2(s0):
In particular, this gives a matching of KLV polynomials for
GL(n;C) and U(p; q).
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.
Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G.
For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint.
Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)dual$ geometry of G_(C) orbits on XR:
Where does all this come from?
Suppose G is a complex algebraic group de�ned over R.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example
GL(n;R) if n is odd; and
GL(n;R) and GL(n=2;H) if n is even.
Simplify: G adjoint. Then main result of ABV
Mi
KHC(Gi)�dual$ geometry of G_(C) orbits on XR;�:
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example�
GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)dual$ geometry of G_(C) orbits on XF :
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.
Fix an inner class of real forms fG1; : : : ; Gkg of G. For example�GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)dual$ geometry of G_(C) orbits on XF :
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.Fix an inner class of real forms fG1; : : : ; Gkg of G.
For example
�GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)dual$ geometry of G_(C) orbits on XF :
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example�
GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)dual$ geometry of G_(C) orbits on XF :
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example�
GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)dual$ geometry of G_(C) orbits on XF :
Where does all this come from?
Suppose G is a an adjoint algebraic group de�ned over F = Qp.Fix an inner class of real forms fG1; : : : ; Gkg of G. For example�
GL(n=d;Ed)
���� djn�
Then Deligne-Langlands-Lusztig:
Mi
KUN (Gi)�dual$ geometry of G_(C) orbits on XF;�:
Where does all this come from?
Write out the spaces. If � is real, then there is a natural map
XF;� �! XR;�:
In the case of G = GL(n), this unravels (on the level of orbits)to give the map
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Where does all this come from?
Write out the spaces. If � is real, then there is a natural map
XF;� �! XR;�:
In the case of G = GL(n), this unravels (on the level of orbits)to give the map
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Where does all this come from?
Write out the spaces. If � is real, then there is a natural map
XF;� �! XR;�:
In the case of G = GL(n), this unravels (on the level of orbits)to give the map
�2 : MS(�) �!a
p+q=n
(GL(p;C)�GL(q;C))nBn:
Generalizations?
If G is simple, adjoint, and classical, one may unravel thenatural map XF;� �! XR;� in much the same way.
Find an anologous matching of KLV polynomials for UN (G=F )and HC(G=R) (and a weaker one for HC(G=C)).
Generalizations?
If G is simple, adjoint, and classical, one may unravel thenatural map XF;� �! XR;� in much the same way.
Find an anologous matching of KLV polynomials for UN (G=F )and HC(G=R)
(and a weaker one for HC(G=C)).
Generalizations?
If G is simple, adjoint, and classical, one may unravel thenatural map XF;� �! XR;� in much the same way.
Find an anologous matching of KLV polynomials for UN (G=F )and HC(G=R) (and a weaker one for HC(G=C)).
Generalizations?
If G is simple, adjoint, and exceptional, the natural mapXF;� �! XR;� is less well behaved.
(Two orbits can collapse toone, for instance.)
Generalizations?
If G is simple, adjoint, and exceptional, the natural mapXF;� �! XR;� is less well behaved. (Two orbits can collapse toone, for instance.)
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Back to the nice case of GL(n)
The matching of KLV polynomials implies there are nicerelationships between character formuals for
1 Harish-Chandra modules for GL(n;C);
2 Unipotent representations of GL(n;Qp); and
3 Harish Chandra modules for GL(n;R).
Are there functors explaining these relationships?
Existence of Functors?
Since we have maps
�1 : XF;� �! XC;�
�2 : XF;� �! XR;�
we anticipate functors
HC(GL(n;C))� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
Existence of Functors?
Since we have maps
�1 : XF;� �! XC;�
�2 : XF;� �! XR;�
we anticipate functors
HC(GL(n;C))� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
Existence of Functors?
Since we have maps
�1 : XF;� �! XC;�
�2 : XF;� �! XR;�
we anticipate functors
HC(GL(n;C))� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
Existence of Functors?
Since we have maps
�1 : XF;� �! XC;�
�2 : XF;� �! XR;�
we anticipate functors
O� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
Existence of Functors?
We expect that the functors
O� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
should take standard modules to standard modules (or zero)and irreducibles to irreducibles (or zero).
See Dan Ciubotaru's talk.
Existence of Functors?
We expect that the functors
O� �! UN (GL(n;Qp))
HC(GL(n;R))� �! UN (GL(n;Qp))
should take standard modules to standard modules (or zero)and irreducibles to irreducibles (or zero).
See Dan Ciubotaru's talk.
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