Affine flag varieties and quantum symmetric pairs, II. Multiplication formula

0.46 MB·

by  Zhaobing Fan

AFFINE FLAG VARIETIES AND QUANTUM SYMMETRIC PAIRS, II.
MULTIPLICATION FORMULA
ZHAOBING FAN AND YIQIANG LI
7
1
0 Abstract. We establish a multiplication formula for a tridiagonalstandard basis element
2
in the idempotented coideal subalgebras of quantum affine gl arising from the geometry
n
n of affine partial flag varieties of type C. We apply this formula to obtain the stabilization
Ja algebras K˙ cn, K˙ nı, K˙ ın and K˙ ıηı, which are idempotented coideal subalgebras of quantum
3 affine gln. The symmetry in the formula leads to an isomorphism of the idempotented
2 coideal subalgebras K˙ nı and K˙ ın with compatible monomial, standard and canonical bases.

]
A
Q
Contents
.
h
at Introduction 2
m 0.1. Overview 2
[ 0.2. Main results 2

1 0.3. The organization 3
v 0.4. Acknowledgement 4
8
1. Recollections from [FLLLWa] 4
4
3 1.1. The space of affine flags of type C as symplectic lattice chains 4
6
1.2. Parametrization matrices 5
0
. 1.3. The convolution algebra Sc 6
1 n,d
1.4. Three variants 6
0
7 2. Multiplication formula for tridiagonal standard basis elements 7
1
2.1. The formula 7
:
v 2.2. Toward a proof I: type A counting 9
i
X 2.3. Toward a proof II: type C counting 16
r 2.4. Step 1 of the proof: the piece Z 19
a S,T
2.5. Step 2 of the proof: counting Z 21
S,T
3. The quantum group K˙ c via the multiplication formula 25
n
3.1. A monomial basis of the convolution algebras 25
3.2. Monomial basis and canonical basis of K˙ c 28
n
3.3. Homomorphism from K˙ c to Sc 30
n n,d
4. The algebras K˙ ı, K˙ ı and K˙ ıı 31
n n η
4.1. Monomial and canonical bases for Sı and K˙ ı 31
n,d n
4.2. The algebras of type ı 34
Date: January 24, 2017.
2010 Mathematics Subject Classification. 17B37,20G25, 14F43.
Key words and phrases. Quantum affine gl , coideal subalgebra, canonical basis, multiplication formula,
n
affine flag variety, convolution algebra.
1
2 Z. FAN AND Y.LI
4.3. Isomorphisms between the types ı and ı 34
4.4. Monomial and canonical bases for algebras of type ıı 36
References 37
Introduction
0.1. Overview. It is well-known that the convolution algebras of flag varieties provide a
natural geometric model for Hecke algebras. As an important feature from this model,
the Kazhdan-Lusztig bases [KL79] of the Hecke algebras can be interpreted as intersection
cohomology complexes therein and from which one can deduce nontrivial properties such as
positivity.
Later, Beilinson, Lusztig and MacPherson [BLM90] observed that partial flag varieties
of type A is a geometric setting for quantum gl . For other classical partial flag varieties,
n
it is only known recently that they are governed by coideal subalgebras of quantum gl
n
([BKLW14, BLW14]).
There is an affinization of the above results making a connection between affine flag va-
rieties of type A and C and quantum affine gl /sl [GV93, Lu99, Lu00, SV00, Mc12] and
n n
their coideal subalgebras ([FLLLWa]).
More precisely, we provide a description of the convolution algebra of affine partial flag
varieties of type C in the work [FLLLWa]. Among others, we show that it is controlled by
certain coideal subalgebra of quantum affine gl and in turn provides canonical bases for the
n
latter algebras. As a crucial ingredient, we show that the convolution algebra admits a set
of bar-invariant multiplicative generators, denoted by f therein, parametrized by certain
A
tridiagonal matrices. Roughly speaking, the bar-invariant basis f corresponds to a product
A
of Chevalley generators in higher-rank convolution algebras in a specific order. In particular,
one can derive a multiplication formula for f by repetitively applying the the multiplication
A
formulas for Chevalley generators, which turns out to be simple. This leads to a construction
of the idempotented version of the associated coideal subalgebras of quantum affine gl and
n
their canonical bases after a suitable stabilization following [BLM90] and [FL14].
There is yet another natural set of multiplicative generators consisting of the tridiagonal
standard basis elements [A], which is parametrized by the same set of tridiagonal matrices
for the generators f . The transition matrix of the two bases turns out to be unitriangular.
A
The purposeofthis paperisto establish amultiplication formula forthetridiagonal standard
basis elements [A], which then provides a direct construction of the affine coideal subalgebras
of quantum affine gl and their canonical bases in [FLLLWa] with some further new and
n
interesting symmetries.
0.2. Main results. Sinceweknowessentiallythemultiplicationformulaforf , thecomplex-
A
ity to establish the multiplication formula for a tridiagonal basis element [A] is incorporated
in the transition matrix of the two bases. which ends up producing coefficients involving
terms (vm 1) for various m. The latter terms vanish in finite types or affine type A since
−
f and [A] coincide for the necessary matrices in these cases and hence the transition matrix
A
is the identity matrix. Follows is a rough form of the multiplication formulas, and we refer
to (24), (76) and (78) for unexplained notations.
MULTIPLICATION FORMULA 3
Theorem A (Theorem 3.2). Let α = (α ) NZ such that α = α for all i Z. If
i i∈Z i i+n
∈ ∈
A,B Ξ satisfy that co(B) = ro(A) and B α Ei,i+1 is diagonal, then we have
∈ n,d − 1≤i≤n i θ
[B] [A] = vhS,Tn(S,PT) AS,T [A ],
∗ S S,T
S,T (cid:20) (cid:21)b
X
where the sum runs over all S,T Θ subject to Condition (20), ro(S) = α and ro(T) = αJ,
n
ˇ ∈
A T +T Θ , and A Ξ .
n S,T n,d
− ∈ ∈
In many respects, the way we derive this formula has many similarities to, and is partially
based on, the one on quantum affine gl constructed in [FL15]. Note that the latter formula
n
is first discovered in [DF13] via an Hecke-algebraic approach. Besides the new construction
in [FL15], the identification of the affine type-A analogue of f and [A] for A bidiagonal
A
indeed gives an easier way to deduce this formula.
Even though the multiplication formula is rather involved, we are able to obtain sufficient
conditions on when a leading term with coefficient 1 is produced under such multiplication,
which allow us to observe a BLM-type stability property as d goes to . The stabilization
also allows us to formulate a limit algebra K˙ c for the family of the ∞convolution algebras
n
Sc . The algorithm for the construction of a monomial basis for Sc leads to an
{ n,d}d≥1 n,d
algorithm for a monomial basis for K˙ c.
n
Theorem B (Theorem 3.12). The algebra K˙ c admits a monomial basis and a canonical
n
basis.
One can define another algebra Kc so that K˙ c is identified with the idempotented version
n n
of a coideal subalgebra of quantum affine gl as is shown in [FLLLWa].
n
Similarly, the other families of convolution algebras Sı , Sı , and Sıı admit
{ n,d}d { n,d}d { η,d}d
similar stabilizations which lead to limit algebras K˙ ı,K˙ ı and K˙ ıı, respectively. We also
n n η
establish the counterparts of Theorem B for the algebras K˙ ı,K˙ ı and K˙ ıı, which require
n n η
some additional work (it follows a strategy similar to [FL14]). Despite quite complicated,
the structure constants in the multiplication formula in Theorem A manifest remarkable
symmetries, reflecting the shift by half-period in parametrization matrices, which allow us
to show that
Theorem C. There is an isomorphism K˙ ı = K˙ ı with compatible monomial, standard and
n ∼ n
canonical bases.
Simultaneously, a Hecke-algebraic approach has been developed in a companion paper
[FLLLWb], which reproduces most of the main results of this paper in different forms. In
light of a result in [FLLLWb], the algebras denoted by the same notations K˙ c, K˙ ı, K˙ ı and
n n n
K˙ ıı in the paper and [FLLLWa, FLLLWb] are isomorphic and they are idempotent versions
η
of coideal subalgebras of quantum affine gl . Having the three approaches available indicate
n
the rich structures in these algebras arising from geometry.
0.3. The organization. In Section 1, we recall the setting from [FLLLWa] on the convolu-
tion algebras Sc , Sı , Sı and Sıı .
n,d n,d n,d η,d
In Section 2, we obtaina multiplication formula for a tridiagonal standard basis element in
the convolution algebra Sc . The proof is rather involved taking up the whole long section,
n,d
4 Z. FAN AND Y.LI
and the formula is the key to the remaining sections on the structures of Sc which lead to
n,d
the construction of the limit algebra K˙ c.
n
In Section 3, we obtain a monomial basis for the convolution algebra Sc based on the
n,d
multiplication formula obtained in Section 2. In particular, it follows that the standard
basis elements parametrized by tridiagonal matrices form a generating set for the algebra
Sc . This multiplication formula admits a remarkable stabilization property which allows
n,d
us to construct a limit algebra K˙ c for the family of convolution algebras Sc . We then
n { n,d}d
construct amonomialbasis andcanonical basisforK˙ c, aswell asasurjective homomorphism
n
from K˙ c to Sc .
n n,d
In Section 4, we adapt the multiplication formula from Section 2 for the other variants
of convolution algebras: Sı , Sı , Sıı . This then allows us to construct the corresponding
n,d n,d η,d
limit algebras K˙ ı, K˙ ı, and K˙ ıı, respectively. Monomial bases and canonical bases for these
n n η
algebras are constructed.
0.4. Acknowledgement. ThepapergrowsoutfromtheProjectFLLLW[FLLLWa,FLLLWb]
and we thank our collaborators, especially Weiqiang Wang, for fruitful collaborations. We
also thank our collaborators in the Project FLLLW for cross-checking our multiplication
formula and are grateful for Weiqiang Wang for tuning up an earlier version of this article.
Z. Fan is partially supported by the Fundamental Research Funds for the central universities
(grant No. GK2110260131) and the NSF of China (grant No. 11671108).
1. Recollections from [FLLLWa]
In this section, we recall the setting from [FLLLWa] and fix some notations. There is no
new result in this section.
1.1. The space of affine flags of type C as symplectic lattice chains. Let N =
0,1,2,... and Z = 0, 1, 2,... . For a Z and b N, we define
{ } { ± ± } ∈ ∈
a v2(a−i+1) 1 a
(1) = − , and [a] = .
b v2i 1 1
(cid:20) (cid:21) 1≤i≤b − (cid:20) (cid:21)
Y
Let k be a finite field of q elements. Let F = k((ε)) be the field of formal Laurent series
over k and o = k[[ε]] the ring of formal power series. Let V be an F-vector space. A lattice
in V is a free o-module such that F = V.
o
L L⊗
Assume further that V is equipped with a non-degenerate symplectic form ( , ). Hence
− −
V is even dimensional, say 2d. Let Sp(V) be the group of isometries with respect to the
form ( , ). For any lattice V, we set # = v V (v, ) o . Then the space
− − L ∈ L { ∈ | L ⊆ }
# is a lattice of V such that ( #)# = . The # operation enjoys the following properties
L L L
that we shall use freely later on. For any two lattices , ′ of V, ( + ′)# = # ′#
L L L L L ∩ L
and ( ′)# = # + ′#. We are interested in the symplectic lattices, which are those
L ∩ L L L
homothetic to a lattice Λ such that εΛ Λ# Λ.
⊆ ⊆
Once and for all, we fix an even number
n = 2r +2, for some r N.
∈
MULTIPLICATION FORMULA 5
Let c be the set of all chains L = (L ) of symplectic lattices in V subject to the
Xn,d i i∈Z
following conditions.
L L , L = L , L# = L , z Z.
z ⊆ z+1 z z+n z −z−1 ∀ ∈
The group Sp(V) acts naturally on c . Let
Xn,d
Λc = λ = (λ ) NZ λ = λ = λ , λ = 2d,λ ,λ 2N .
n,d { i i∈Z ∈ | i i+n −i−1 i 0 r+1 ∈ }
1≤i≤n
X
For each λ Λc , we set
∈ n,d
(2) c (λ) = L c dim L /L = λ i Z .
Xn,d { ∈ Xn,d| k i i−1 i ∀ ∈ }
Then we have
(3) Xnc,d = ⊔λ∈Λcn,dXnc,d(λ),
as the union of Sp(V)-orbits in c . From the analysis of [Sa99], [H99] (see also [FLLLWa]),
Xn,d
we see that c (λ) can be naturally identified with the homogeneous space Sp(V)/P where
Xn,d
P is a certain parahoric subgroup. Hence c is a local model of the ind-variety of affine
Xn,d
partial flags of type C over k.
1.2. Parametrization matrices. The Sp(V)-action on c extends diagonally on the
Xn,d
product c c . We recall a parametrization of Sp(V)-orbits in c c . Let Θ be
Xn,d×Xn,d Xn,d×Xn,d n,d
the set of all matrices A = (a ) with entries in N such that
ij i,j∈Z
i0+n−1
a = a , i,j Z; a = d, i Z.
ij i+n,j+n ij 0
∀ ∈ ∀ ∈
Xi=i0 Xj∈Z
To a matrix A Θ , we associate its row/column vector ro(A) = (ro(A) ) and co(A) =
n,d i i∈Z
∈
(co(A) ) by
i i∈Z
ro(A) = a , co(A) = a , i,j Z.
i ij j ij
∀ ∈
j∈Z i∈Z
X X
Let
(4) cΞ = A Θ a = a = a , i,j Z,a ,a 2N .
n,d n,2d ij −i,−j n+i,n+j 00 r+1,r+1
{ ∈ | ∀ ∈ ∈ }
We set Eij to be the matrix whose entry at (k,l) is 1 if (k,l) (i,j) mod n and zero
≡
otherwise. We also set
Eij = Eij +E−i,−j.
θ
For later use, let
(5) Ξ = A+E0,0 +Er+1,r+1 A cΞ .
n,d n,d
{ | ∈ }
Clearly, there is a bijection
(6) cΞ = Ξ .
n,d ∼ n,d
To a pair (L,L′) c c , we can attach a matrix A cΞ whose (i,j)-th entry is
∈ Xn,d ×Xn,d ∈ n,d
given by
a = dim L L′/(L L′ +L L′ ), i,j Z.
ij k i ∩ j i−1 ∩ j i ∩ j−1 ∀ ∈
6 Z. FAN AND Y.LI
From [FLLLWa], the correspondence (L,L′) A yields a bijection
7→
(7) Sp(V) c c cΞ .
\Xn,d ×Xn,d → n,d
We shall denote for the Sp(V)-orbit parametrized by A. Let e be the characteristic
A A
O
function of . Under the isomorphism (6), we can also parametrize the Sp(V)-orbits in
A
O
c c by the set Ξ . The notation e also makes sense for a matrix A Ξ .
Xn,d ×Xn,d n,d A ∈ n,d
1.3. The convolution algebra Sc . Let Sc be the vector space over Q(v) spanned by
n,d n,d
the elements e for A cΞ . There is a multiplication on Sc defined by
A ∈ n,d n,d
(8) e e = gC (v),e , gC (v) Z[v,v−1],
A ∗ B A,B C A,B ∈
C∈XcΞn,d
where the specialization of the polynomial gC (v) at √q is given by
A,B
(9) gC (√q) = # L˜ c (L,L˜) ,(L˜,L′)
A,B { ∈ Xn,d| ∈ OA ∈ OB}
for any fixed pair (L,L′) . It is known that gC = 0 for all but finitely many C. The
∈ OC A,B
algebra Sc is the convolution algebra on c and is called the Schur algebra in [FLLLWa].
n,d Xn,d
To each A cΞ , we set
n,d
∈
1
dc = a a + a + a ,
A 2  ij kl ij ij
1≤i≤n i≥0>j i≥r+1>j
X X X
i≥k,j<l 
 
and
[A] = v−dcAeA.
Clearly, the various elements [A] form a basis for Sc , called the standard basis of Sc .
n,d n,d
Define a partial order on cΞ by
alg n,d
≤
A B if and only if a b , i < j.
alg kl kl
≤ ≤ ∀
k≤i,l≥j k≤i,l≥j
X X
We write A < B if further A = B. By “lower terms (than [B])”, we refer to the terms [A]
alg
6
with A < B, ro(A) = ro(B) and co(A) = co(B).
alg
There is a bar operator¯on Sc satisfying [A] = [A]+lower term. To each matrix A, there
n,d
is a unique element A in Sc that is bar-invariant and A [A] v−1Z[v−1][B].
{ } n,d { }− ∈ B<algA
The A s clearly is a basis of Sc , called the canonical basis of Sc .
{ } n,d n,d P
1.4. Three variants. We now set
n = n 1 = 2r+1 for r 1.
− ≥
We consider the subset Ξı of Ξ defined by
n,d n,d
(10) Ξı = A Ξ a = δ ,a = δ , i,j Z .
n,d { ∈ n,d| r+1,j r+1,j i,r+1 i,r+1 ∀ ∈ }
We define an idempotent in Sc by
n,d
(11) j = [A],
r
X
MULTIPLICATION FORMULA 7
where the sum runs over all diagonal matrices in Ξı . We define the subalgebra Sı of Sc
n,d n,d n,d
by
(12) Sı = j Sc j .
n,d r n,d r
It is known from [FLLLWa] that Sı inherits from Sc a basis of characteristic functions
n,d n,d
e , a standard basis [A] and a canonical basis A parametrized by the set Ξı .
A { } n,d
On the other hand, we consider the set Ξı of Ξ given by
n,d n,d
(13) Ξı = A Ξ a = δ ,a = δ , i,j Z .
n,d { ∈ n,d| 0,j 0,j i,0 i,0 ∀ ∈ }
We define j for Ξı in exactly the same way as j in (11) and we set
0 n,d r
(14) Sı = j Sc j .
n,d 0 n,d 0
It is known again from [FLLLWa] that Sı inherits from Sc a basis of characteristic func-
n,d n,d
tions e , a standard basis [A] and a canonical basis A parametrized by the set Ξı .
A { } n,d
Finally, we set η = n 1 = 2r and
−
(15) Ξıı = Ξı Ξıı .
η,d n,d ∩ n,d
We define
(16) Sıı = Sı Sı ,
η,d n,d ∩ n,d
and we know from [FLLLWa] that Sıı inherits from Sc a basis of characteristic functions
η,d n,d
e , a standard basis [A] and a canonical basis A parametrized by the set Ξıı .
A { } η,d
2. Multiplication formula for tridiagonal standard basis elements
In this section, we obtain a multiplication formula in the convolution algebra Sc for a
n,d
tridiagonal standard basis element. The formula is the key to the remaining sections on the
structures of Sc which lead to the construction of the limit algebra K˙ c. The proof of the
n,d
formula is rather involved, taking up the whole section.
Note that the formulation for the multiplication formulas in this section use the index set
cΞ in (4) for bases of Sc , as it is most convenient to use cΞ in its proof. It will be
n,d n,d n,d
reformulated in terms of Ξ in Section 3.
n,d
2.1. The formula. Let n = 2r+2 with r 0. We denote
≥
Θ = Θ
n n,d
(17) d≥0
G
= (s ) Mat (N) s = s , i,j Z .
ij Z×Z ij i+n,j+n
{ ∈ ∀ ∈ }
We also denote (cid:12)
(cid:12)
(18) Θ = (s ) Mat (Z) s 0 ( i = j),s = s , i,j Z .
n ij Z×Z ij ij i+n,j+n
{ ∈ ≥ ∀ 6 ∀ ∈ }
We define a (row shift) bijectionˇ:(cid:12)Θn Θn by sending S = (sij) to
e (cid:12) →
Sˇ = (sˇ ) , sˇ = s ( i,j Z).
ij i,j∈Z ij i−1,j
∀ ∈
Given A,S,T Θ , we set
n
∈
ˇ ˇ
(19) A = A+S T (S T).
S,T
− − −
8 Z. FAN AND Y.LI
If we write A = (a′ ) , then a′ = a′ for all i,j Z. But a′ may be negative, so
S,T ij i,j∈Z ij i+n,j+n ∈ ij
A is not in Θ in general. Consider the following subset Ξ of Θ :
S,T n n n
Ξ = A = (a ) Θ a = a , i,j Z .
n ij i,j∈Z n ij −i,−j
{ ∈ ∀ ∈ }
Lemma 2.1. Let A Ξ and S,T Θ be suc(cid:12)h that A Θ . Then we have A Ξ
∈ n ∈ n (cid:12) S,T ∈ n S,T ∈ n
if and only if S,T satisfy
(20) s +s = t +t , i,j Z.
ij −i−1,−j ij −i−1,−j
∀ ∈
Proof. By substituting i with i 1 in Condition (20), we have s t = t s .
i−1,j i−1,j −i,−j −i,−j
− − −
Denoting A = (a′ ), we have thus obtained
S,T ij
(21) a′ = a +(s +s ) (t +t ).
ij ij ij −i,−j − ij −i,−j
It now follows that a′ = a′ , whence A Θ .
ij −i,−j S,T ∈ n (cid:3)
The above argument can be reversed to establish the opposite direction.
Let
= r 1 ( r 1, ) [ r, 1] Z 0 ( ,0) Z Z.
J {− − }× − − ∞ − − × { }× −∞ ⊆ ×
Recall the q(cid:0)uantum v-binomials from(cid:1)G(1)(cid:0). For S = (s (cid:1))Gand(cid:0) A = (a′ ), w(cid:1)e set
ij S,T ij
A a′ a′ s [a′ 2k]
(22) S,T = ij ij − ij ii − .
S s s [k +1]
(cid:20) (cid:21)c (i,j)∈J (cid:20) ij(cid:21)(cid:20) −i,−j (cid:21) i=0,−r−1
Y Y
0≤k≤sii−1
Given three sequences α = (α ) ,β = (β ) and γ = (γ ) in NZ such that their
i i∈Z i i∈Z i i∈Z
entries are all zeros except at finitely many places, we define
(23)
[β ]!
n(α,β,γ) = v2(Pi,j,l:j>lσijαl+Piσiiαi+Pi>k,j<lσijσkl) i∈Z i (v2αj v2l),
[σ ]! −
σ=X(σij) Qi,j∈Z ij Yj∈Z
0≤l≤αj−σjj−1
Q
where the sum runs over all upper triangular matrices σ = (σ ) Mat (N) such that
ij Z×Z
∈
ro(σ) = β and co(σ) = γ.
Given a sequence a = (a ) , we define the sequence aJ whose i-th entry is a for all
i i∈Z −i
i Z. We set
∈
(24) n(S,T) = n(S ,(S )J,T ),
i −i−1 i
0≤i≤r
Y
where S and T are the i-th row vectors of S and T, respectively, and n(α,β,γ) is in (23).
i i
To α = (α ) ZZ, we set
i i∈Z
∈
(25) α# = (α#) , where α# = α , i Z.
i i∈Z i −i−1 ∀ ∈
Recall the subset cΞ of Ξ from (4). Given A cΞ and S,T Θ , we denote
n,d n n,d n
∈ ∈
s (s 1)
(26) ξc = (a′ s )s + s s ij ij − .
A,S,T ij − ij il − −i,−j il − 2
−r−X1≤i≤r (cid:16)−rX≤i≤−1 i=−Xr−1,0(cid:17) i=−Xr−1,0
j>l j>l 2i−l>j>l j<i
MULTIPLICATION FORMULA 9
We can now state the general multiplication formulas for the convolution algebra Sc ; for
n,d
A
notations A , S,T and n(S,T) see (19), (22) and (24).
S,T S
(cid:20) (cid:21)c
Theorem 2.2. Let α = (α ) NZ such that α = α for all i Z. If A,B cΞ
i i∈Z i i+n n,d
∈ ∈ ∈
satisfy co(B) = ro(A) and B α Ei,i+1 is diagonal, then we have
− 1≤i≤n i θ
(27) eB ∗eAP= v2ξAc,S,T n(S,T) ASS,T eAS,T,
S,T (cid:20) (cid:21)c
X
where the sum runs over S,T Θ subject to Condition (20), ro(S) = α, ro(T) = α#,
n
∈
A T +Tˇ Θ , A cΞ .
n S,T n,d
− ∈ ∈
We make some remarks before providing the proof of this theorem.
Remark 2.3. ThematrixA dependsonlyonA,S ,T fori [0,r]. Indeed, bysymmetries
S,T i i
∈
at (0,0) and (r +1,r +1), the matrix A is completely determined by the entries a′ for
S,T ij
0 i r +1. Furthermore, for 1 i r, the entry a′ is clearly determined by A and S ,
≤ ≤ ≤ ≤ ij k
T for k [0,r]. For i = 0, we use (21) to get a′ = a + s t (s t ). For
k ∈ 0j 0j 0,j − 0,j − 0,−j − 0,−j
i = r +1, Condition (20) allows us to rewrite a′ = a +s t +(s t ).
r+1,j r+1,j r,−j − r,−j r,j − r,−j
Remark 2.4. If S and T satisfy
(28) s < t
−i−1,−j ij
j≤k j≤k
X X
for some i [0,r] and k, then the structural constant n(S ,SJ ,T ), and hence n(S,T)
∈ i −i−1 i
as well as the coefficient of e in (27), must be zero. This is because the summation
AS,T
in n(S ,SJ ,T ) is taken over upper triangular matrices σ such that ro(σ) = SJ and
i −i−1 i −i−1
co(σ) = T , which is empty if (28) is assumed.
i
Remark 2.5. Note that Conditions (20) and (28) imply that
if s = 0 for some l, then t = s for all j l.
j≥l ij ij −i−1,−j ≥
This allows to reduPce the general multiplication formula to the special cases available for
comparisons. In particular, the general formula (27) is compatible with the formula for
Chevalley generators in [FLLLWa].
The remainder of thissection isdevoted to theproof ofTheorem 2.2. In theproof, we shall
work over a finite field F , and all the quantum numbers and quantum binomial coefficients
q
(defined via the indeterminate v) are understood below at the specialization v = √q (i.e.,
v2 = q).
2.2. Toward a proof I: type A counting. Let V be a finite dimensional vector space over
F . Let us fix a flag W = (0 = W W W = V) of type w = (w ) , i.e.,
q 0 1 m i 1≤i≤m
⊆ ⊆ ··· ⊆
W /W = w for all 1 i m. To a sequece a = (a ) Nm, we set
i i−1 i i 1≤i≤m
| | ≤ ≤ ∈
(29) Ya(W) = U V U Wi U Wi−1 = ai, 1 i m .
{ ⊆ | ∩ |−| ∩ | ∀ ≤ ≤ }
The following lemma can be found(cid:12)in [Sch06, Example 2.4], see also Remark 2.13 for a proof.
(cid:12)
10 Z. FAN AND Y.LI
w
Lemma 2.6. We have #Ya(W) = qPi>kai(wk−ak) i .
1≤i≤m a
i
(cid:20) (cid:21)
Q
Let σ be an upper triangular matrix such that ro(σ) = w. Let t = co(σ) and t(V) be
F
the set of all flags in V of type t. Consider the set
W F
(30) FtW,σ = F ∈ Ft(V) W Fi+∩Wj F = σij,∀1 ≤ i,j ≤ m .
n (cid:12) (cid:12) i−1 ∩ j i ∩ j−1(cid:12) o
(cid:12) (cid:12)
(cid:12)
Here W = (W ) is a fixed(cid:12)fl(cid:12)ag of type w. (cid:12)
i 1≤i≤m (cid:12) (cid:12)
Lemma 2.7. The cardinality of W is given by
Ft,σ
[w ]!
(31) #FtW,σ = qPi>k,j<lσijσkl [σ ]![σ ]i! [σ ]!.
ii i,i+1 im
1≤i≤m ···
Y
Proof. We fix j 1 steps F , ,F such that Wi∩Fj′ = σ for all 1 i m
− 1 ··· j−1 Wi−1∩Fj′+Wi∩Fj′−1 i,j′ ≤ ≤
and 1 j′ j 1. We want to determine the (cid:12)number of choices(cid:12) of F such that F
j j−1
≤ ≤ − (cid:12) (cid:12) ⊆
F W and Wi∩Fj = σ for all 1 i(cid:12) m. Via the re(cid:12)duction to W /F , this
j ⊆ j Wi−1∩Fj+Wi∩Fj−1 ij ≤ ≤ j j−1
is the same as(cid:12)counting the nu(cid:12)mber of choices of F¯ in W /F such that
(cid:12) (cid:12) j j j−1
(cid:12) (cid:12)
W W
¯ i ¯ i−1
F F = σ , 1 i j.
j j ij
∩ F − ∩ F ∀ ≤ ≤
(cid:12) j−1(cid:12) (cid:12) j−1(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
Note that Wi+Fj−1 (cid:12)Wi−1+Fj−1 (cid:12)= w(cid:12) Wi∩Fj(cid:12)−1 = w σ . By Lemma 2.6, this
Fj−1 − Fj−1 i− Wi−1∩Fj−1 i− j−1≥l il
number is(cid:12)equal to(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) P
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
w σ
qPi,k:i>kσij(wk−Pj≥lσkl) i − j≤i−1 ij .
σ
ij
1≤i≤m(cid:20) P (cid:21)
Y
Taking productover allj andusing w σ = σ , we have provedthelemma. (cid:3)
k− j≥l kl j<l kl
P P
Let
V = (0 = V V V V V = V),
0 1 2 3 4
(32) ⊆ ⊆ ⊆ ⊆
V′ = (0 = V′ V′ V′ V′ = V)
0 ⊆ 1 ⊆ 2 ⊆ ··· ⊆ m
be two partial flags of V. We set
V V′
c = c (V,V′) = i ∩ j , 1 i 3,1 j m.
ij ij V V′ ∀ ≤ ≤ ≤ ≤
(cid:12) i ∩ j−1(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
To a triple (s,t,t′) in Nm, we define(cid:12) (cid:12)
(33) Ys,t,t′ Ys,t,t′(V,V′)
≡