The Brundan-Kleshchev subalgebra and BK-type isomorphsims
aa r X i v : . [ m a t h . R T ] F e b THE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPEISOMORPHISM
FAN KONG AND ZHI-WEI LI*
Abstract.
We use a unified method to give an isomorphism between direct sumsof cyclotomic affine (and degenerate affine) Hecke algebras and cyclotomic BK-subalgebras which are some KLR-type algebras. Introduction
Brundan and Kleshchev [3] gave an explicit isomorphism between blocks of cy-clotimic affine Hecke algebras of symmetric groups and cyclotomic KLR algebras oftype A. But their proof is complicated since the invalidity of some rational functions.On the other hand, Lusztig has showed that there is a natural isomorphism be-tween affine Hecke algebras of all types and their graded versions [7]. One keypoint is the rationalization of affine Hecke algebras. Motivated by Lusztig’s work,we introduce the Lusztig extension of affine Hecke algebras of all types and theBrundan-Kleshchev subalgebra (or BK subalgebra for short) of the Lusztig exten-sion. Surprisingly, we can construct the KLR-type generators of Lusztig extensionsby following Brundan and Kleshchev, and then fast give an isomorphism betweendirect sums of blocks of cyclotomic affine (and degenerate affine) Hecke algebras andcyclotomic BK subalgebras.In order to give an overview of our main results in this article, let D be the directedDynkin diagram of a fixed Weyl group W with vertex set [ n ] := { , , · · · , n } . We Date : February 17, 2021.*Corresponding author. FAN KONG AND ZHI-WEI LI use ( a rs ) r,s ∈ [ n ] to denote the Cartan matrix of D : a rs := r = s ;0 if r / − s ; − r − s , or r h s , or r h s ; − r i s ; − r i s . Here the symbol r / − s indicates that r is not connected with s by edges; r − s indicates that r is connected with s by a single edge; r i s indicates that r isconnected with s by a double edge and there is an arrow from r to s ; and r i s indicates that r is connected with s by a triple edge and there is an arrow from r to s . Let I n be the set of n -tuples of the abelian group I = Z /e Z ( e ≥ e = 1).The Weyl group W acts on the left on I n via (2.6). We fix a W -orbit C of I n anda system { ǫ ( i ) | i ∈ C} of mutually orthogonal idempotents. Fix a ground field k .We consider the BK subalgebra ˜ L generated by { y , · · · , y n , ψ , · · · , ψ n } ∪ { ǫ ( i ) | i ∈ C} ∪ { f − | f ∈ k [ y , · · · , y n ] with f (0) = 0 } and some KLR-type basis relations (1)-(9) of Theorem 3.40.Let H be the degenerate affine Hecke algebra generated by x , · · · , x n , t , · · · , t n and relations (3.1)-(3.7). From now on, we fix Λ = (Λ i ) i ∈ I ∈ N I (we follow theconvention that N = { , , , · · · } ) with P i ∈ I Λ i < ∞ . Let ˜ L (Λ) = ˜ L / h y Λ i ǫ ( i ) | i ∈Ci and and H (Λ) = H / h Q i ∈ I ( x − i ) Λ i i be the corresponding cyclotomic quotients.The following is our first main result in this article for the degenerate affine Heckealgebra: Theorem 3.48
There is an isomorphism of algebras ˜ L (Λ) ∼ = H (Λ) e ( C ) . Next, we give a similar construction for the non-degenerate affine Hecke algebra.Fix q ∈ k , q = 0 ,
1. We consider the
BK subalgebra ˜ L q generated by { Y , · · · , Y n , Ψ , · · · , Ψ n } ∪ { ǫ ( i ) | i ∈ C} ∪ { f − | f ∈ k [ Y , · · · , Y n ] with f (0) = 0 } and some KLR-type basis relations (1)-(10) of Theorem 4.38. HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 3
Let H q be the non-degenerate affine Hecke algebra generated by X , · · · , X n ,T , · · · , T n and relations (4.1)-(4.7). Let ˜ L q (Λ) = ˜ L / h Y Λ i ǫ ( i ) | i ∈ Ci and and H q (Λ) = H / h Q i ∈ I ( X − i ) Λ i i be the corresponding cyclotomic quotients.The following is our second main result in this paper for the non-degenerate affineHecke algebra: Theorem 4.45
There is an isomorphism of algebras ˜ L q (Λ) ∼ = H q (Λ) e ( C ) . We now sketch the contents of the paper. In section 2, we give the group actionof W on the n -tuples I n which will be used as an index set in the whole article.In Section 3, we first review the degenerate affine Hecke algebras and recall theirBernstein-Zelevinski basis. We then introduce the Lusztig extension of degenerateaffine Hecke algebras and give their KLR-type generators. Our first main resultTheorem 3.48 in this paper are then proved after introducing the BK subalgebras.In Section 4, we consider the non-degenerate case in a similar way. The last sectionis aimed to give the general and unified definition for the KLR type algebras in thetwo cases above. 2. Preliminaries
The divided difference operator of polynomial rings.
Denote by [ n ] = { , , · · · , n } . We view the Weyl group W as being generated as a Coxeter group withgenerators σ r , r = 1 , , · · · , n . Let k [ x ] be the polynomial ring with indeterminates { x r | r ∈ [ n ] } over the field k . There is an action of W from the left on k [ x ] (by thering automorphism) by defining σ r ( x s ) = x s − a sr x r . (2.1)for r, s ∈ [ n ]Using the W -action above, we define the divided difference operators ∂ r on k [ x ]for all r ∈ [ n ] as ∂ r ( f ) = σ r ( f ) − fx r . By straightforward calculations, the divided difference operators satisfy the Leibnizrule ∂ r ( f g ) = ∂ r ( f ) g + σ r ( f ) ∂ r ( g ) , for f, g ∈ k [ x ] and r ∈ [ n ], and the relations σ r ( ∂ r ( f )) = ∂ r ( f ) , ∂ r ( σ r ( f )) = − ∂ r ( f ) FAN KONG AND ZHI-WEI LI
Let k ( x ) be the corresponding rational function field, then the W -action on k [ x ]above can be extended to an action w : fg w ( f ) w ( g ) of W on k ( x ) (by the field auto-morphism). This means that the action of the divided difference operators on k [ x ]also extends to operators on k ( x ).Let F be the quotient field of the subalgebra Z = { f ∈ k [ x ] | w ( f ) = f for every w ∈ W} of W -invariants. By [7, 3.12 (a)], there is a natural k -algebra isomorphism k [ x ] ⊗ Z F → k ( x ) , f ⊗ g f g. (2.2)2.2. The Demazure operator of Laurent polynomials rings.
Let k [ X ± ] bethe Laurent polynomial ring in the indeterminates { X r | r ∈ [ n ] } . There is an actionof W from the left on k [ X ± ] (by the ring automorphism) by defining σ r ( X s ) = X s X − a sr r . (2.3)for all r, s ∈ [ n ]. Using the W -action above, we define Demazure operators D r on k [ X ± ] for all r ∈ [ n ] as D r ( f ) := σ r ( f ) − f − X r . (2.4)It is well-known that the Demazure operators on k [ X ± ] also satisfy the Leibniz ruleD r ( f g ) = D r ( f ) g + σ r ( f )D r ( g ) . for f, g ∈ k [ X ± ] and r ∈ [ n ].Let k ( X ) be the corresponding rational function field, then the action of W on k [ X ± ] extends to an action w : fg w ( f ) w ( g ) of W on k ( X ) (by the field automorphism).This means that the action of the Demazure operators on k [ X ± ] also extends tooperators on k ( X ).Let F be the quotient field of the subalgebra Z = { f ∈ k [ X ± ] | w ( f ) = f for every w ∈ W} of W -invariants. Similar to (2.2), there is a natural k -algebra isomorphism k [ X ± ] ⊗ Z F → k ( X ) , f ⊗ g f g. (2.5) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 5
The numbers game.
Let I be the abelian group Z /e Z ( e ≥ e = 1).For r ∈ [ n ] and i = ( i s ) s ∈ [ n ] ∈ I n , we define the map σ r ( i ) s = i s − a sr i r . (2.6)Then these maps induce an action of W on I n . In fact, this is a special case ofthe numbers game which applies a combinatorial model of the Coxeter groups [1,Theorem 4.3.1(ii)]. Lemma 2.7.
For r, s ∈ [ n ] , there holds that (1) σ r = 1 . (2) σ r σ s = σ s σ r , if r / − s . (3) σ r σ s σ r = σ s σ r σ s , if r − s . (4) ( σ r σ s ) = ( σ s σ r ) , if r i s . (5) ( σ r σ s ) = ( σ s σ r ) , if r i s .Proof. The assertions (1)-(2) can be proved directly from the construction of σ r .The assertions (3)-(5) follow from the following commutative diagrams: i r i s i r + i s − i s − i r i r + i s − i r − i s i r i s − i r − i s − i s − i r σ s σ r σ r σ s σ s σ r i i r i s i i r + 2 i s − i s i − i r i r + i s i − i r − i s i r + i s i i r + 2 i s − i r − i s i i r − i r − i s i − i r − i s i s i − i r − i s σ s σ r σ r σ s σ s σ r σ r σ s i i r i s i i r + 3 i s − i s i − i r i r + i s i i r + 3 i s − i r − i s i − i r − i s i r + 2 i s i − i r − i s i r + 2 i s i i r + 3 i s − i r − i s i − i r − i s i r + i s i i r + 3 i s − i r − i s i i r − i r − i s i − i r − i s i s i − i r − i s σ s σ r σ r σ s σ s σ r σ r σ s σ r σ s (cid:3) In the sequel, for simplicity, we will write i s := σ r ( i ) s , i s := σ r σ s ( i ) s , i s := σ r σ s σ r ( i ) s , i s := σ r σ s σ r σ s ( i ) s , i s := σ r σ s σ r σ s σ r ( i ) s .3. The BK subalgebra in degenerate case
Degenerate affine Hecke algebras and their BZ basis.
In [5], Drinfeldintroduced a machinery to define degenerate affine Hecke algebras and gave theconcise form of type A. We notice that Drinfeld’s degenerate affine Hecke algebrais also a quotient of the graded Hecke algebra induced by Lusztig in [7, Proposition4,4]. Following Drinfeld, the degenerate affine Hecke algebra H of W is defined to FAN KONG AND ZHI-WEI LI be the associated unital k -algebra with generators { x r , t r | r ∈ [ n ] } subject to thefollowing relations for all admissible indices: x r x s = x s x r ; (3.1) t r x s = σ r ( x s ) t r + ∂ r ( x s ); (3.2) t r = 1; (3.3) t r t s = t s t r if r / − s ; (3.4) t r t s t r = t s t r t s if r − s ; (3.5)( t r t s ) = ( t s t r ) if r i s ; (3.6)( t r t s ) = ( t s t r ) if r i s . (3.7)For w = σ r σ r · · · σ r m ∈ W a reduced expression we put T w := T r T r · · · T r m .Then T w is a well-defined element in H and the algebra H is a free k [ x ]-module withbasis { T w | w ∈ S n } (the Bernstein-Zelevinski basis) [7, Lemma 3.4].By [7, Proposition 3.11], the center of H is Z . Then H can be seen as a Z -subalgebra (identified with the subspace H ⊗
1) of the Z -algebra H F := H ⊗ Z F. For any s ∈ [ n ] and f ∈ k ( x ), as a consequence of [7, 3.12 (d)], we get t s f = σ s ( f ) t s + ∂ s ( f ) . (3.8)Using the BZ basis of H , there are decompositions [7, 3.12(c)]: H F = ⊕ w ∈W t w k ( x ) = ⊕ w ∈W k ( x ) t w . (3.9)So the rationalization algebra H F is a free k ( x )-module of finite rank with basis { t w | w ∈ W} .3.2. Intertwining elements.
For r ∈ [ n ], we define the intertwining elements κ r in H F as follows: φ r := t r + x − r . The following result is important for us.
Proposition 3.10.
The algebra H F is generated by { x , · · · , x n , φ , · · · , φ n , f − | = f ∈ K [ x ] } subject to the following relations for all admissible r, s : x r x s = x s x r ; (3.11) f f − = f − f = 1 ∀ = f ∈ k [ x ]; (3.12) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 7 φ r x s = σ r ( x s ) φ r ; (3.13) φ r = 1 − x − r ; (3.14) φ r φ s = φ s φ r if r / − s ; (3.15) φ r φ s φ r = φ s φ r φ s if r − s ; (3.16)( φ r φ s ) = ( φ s φ r ) if r i s ; (3.17)( φ r φ s ) = ( φ s φ r ) if r i s . (3.18) Proof.
Since t r = φ r − x − r , thus the elements { x , · · · , x n , φ , · · · , φ n , f − | = f ∈ k [ x ] } are generators of H F by (3.9). The statements (3.11) and (3.12) follow usingthe fact that k ( x ) is a commutative subalgebra of H F . By [7, Proposition 5.2], therelations (3.13)-(3.18) hold. Next we prove the generating set is complete, that is,these relations generate all the relations.Assume A is the k -subalgebra generated by { x , · · · , x n , φ , · · · , φ n , f − | = f ∈ k [ x ] } subject to the relations (3.11)-(3.18). Then there is an obvious surjective k -algebra homomorphism π : A → H F , a a. This maps is k ( x )-linear. So π is in fact a homomorphism of k ( x )-modules. Since H F is a free k ( x )-module of rank |W| with basis { t w | w ∈ W} , so it is enoughto show that A has also rank |W| as a k ( x )-module. In fact, we can define φ w := φ r φ r · · · φ r m if w = σ r σ r · · · σ r m is a reduced expression in W . It is a well-definedelement by the braid relations (3.15)-(3.18). We claim that { φ w | w ∈ W} is a k ( x )-basis of A . In fact, by the relation (3.13), we can express any element a ∈ A in the form a = X w ∈W f w φ w where f w ∈ k ( x ). Since φ w = t w + P u The Lusztig extension of H with respect to E is the k -algebra L which is equal as k -space to the tensor product L := H F ⊗ k E = ⊕ w ∈W , i ∈C φ w k ( x ) ǫ ( i )of the rationalization algebra H F and the semi-simple algebra E . Multiplication isdefined so that H F (identified with the subspace H F ⊗ 1) and E (identified with thesubspace 1 ⊗ E ) are subalgebras of L , and in addition x r ǫ ( i ) = ǫ ( i ) x r , (3.21) φ r ǫ ( i ) = ǫ ( σ r ( i )) φ r . (3.22)It is easy to see that t r ǫ ( i ) = ǫ ( σ r ( i )) t r + x − r ǫ ( σ r ( i )) − x − r ǫ ( i ) . (3.23)We notice that relations (3.20)-(3.23) have been implicitly indicated in [7, Sub-sections 8.7-8.8]. Therefore, the reason for using the notion of Lusztig extension isclear.By Proposition 3.10, the Lusztig extension L has two generating sets, one is { x , · · · , x n , φ , · · · , φ n , f − , ǫ ( i ) | = f ∈ k [ x ] , i ∈ C} , the other one is { x , · · · , x n , t , · · · , t n , f − , ǫ ( i ) | = f ∈ k [ x ] , i ∈ C} . Moreover, the algebra L has decompositions L = ⊕ w ∈W , i ∈C k ( x ) t w ǫ ( i ) = ⊕ w ∈W , i ∈C t w k ( x ) ǫ ( i ) (3.24)and L = ⊕ w ∈W , i ∈C k ( x ) φ w ǫ ( i ) = ⊕ w ∈W , i ∈C φ w k ( x ) ǫ ( i ) . (3.25)3.4. Brundan-Kleshchev auxiliary elements. Recall that [3, (3.12)], for each r ∈ [ n ], Brundan and Kleshchev introduced the element y r := X i ∈C ( x r − i r ) ǫ ( i ) . (3.26)Then y r ǫ ( i ) = ǫ ( i ) y r by 3.21, and y r is a unit in L with y − r = P i ∈C ( x r − i r ) − ǫ ( i ) . Let k [ y ] be the polynomial ring with y := y , y , · · · , y n and k ( y ) be the rationalfunction field. Lemma 3.27. k ( x ) ⊗ k E = k ( y ) ⊗ k E in L as subalgebras. HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 9 Proof. If g ( y ) is a polynomial in k [ y ], then g ( y ) = X i ∈C g ( y ) ǫ ( i ) = X i ∈C g ( x − i , · · · , x n − i n ) ǫ ( i )in L . Therefore, if g ( y ) = 0, then g ( x − i , · · · , x n − i n ) = 0 in K [ x ] for all i ∈ C .Thus g ( y ) − = P i ∈C g ( x − i , · · · , x n − i n ) − ǫ ( i ) exists in L . So k ( y ) ⊗ k E is asubalgebra of L . Combining (3.26) with x r = P i ∈C ( y r + i r ) ǫ ( i ), we know that k ( x ) ⊗ k E = k ( y ) ⊗ k E in L . (cid:3) Following Brundan and Kleshchev, for each r ∈ [ n ], we define the element q r ( i ) = − y r if i r = 0 , i r = 1 , e = 2 , ( y r − − y r ) − if i r = − , e = 2 , (1 − y r ) − if i r = 1 , e = 2 , − ( y r + i r ) − if i r = 0 , ± W acts on the left on k [ y ] since σ r ( y s ) = X i ∈C ( σ r ( X s ) − i s ) ǫ ( σ r ( i )) = y s − a sr y r , for all r, s ∈ [ n ]. For simplicity, we shall write y r := σ s ( y r ) , y r := σ r σ s ( y r ) , y r = σ s σ r σ s ( y r ) , y r = σ r σ s σ r σ s ( y r ) , y r = σ s σ r σ s σ r σ s ( y r ) in the sequel.Similar to the proof of [3, (3.27)-(3.29)], we obtain Lemma 3.28. Let r, s ∈ [ n ] and i ∈ C . Then q r ( i ) σ r ( q r ( σ r ( i ))) = − y r if i r = 0; − ( y r + 2)(1 + y r ) − if i r = 1 , e = 2;( y r − − y r ) − if i r = − , e = 2;(1 − y r ) − if i r = 1 , e = 2;1 − ( y r + i r ) − if i r = 0 , ± . (3.29)Inside L , for each r ∈ [ n ], we set θ r := φ r X i ∈C q − r ( i ) ǫ ( i ) . Using Lemma 3.27 and relations (3.11)-(3.18), we know that the elements θ r havethe following nice properties: FAN KONG AND ZHI-WEI LI θ r ǫ ( i ) = ǫ ( σ r ( i )) θ r ; (3.30) y s θ r = θ r σ r ( y s ); (3.31) θ r θ s = θ s θ r if r / − s ; (3.32) θ r θ s θ r = θ s θ r θ s if r − s ; (3.33)( θ r θ s ) = ( θ s θ r ) if r i s ; (3.34)( θ r θ s ) = ( θ s θ r ) if r i s ; (3.35)and θ r ǫ ( i ) = − y − r ǫ ( i ) if i r = 0 , − y r ǫ ( i ) if i r = 1 , e = 2 ,y r ǫ ( i ) if i r = − , e = 2 , − y r ǫ ( i ) if i r = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . (3.36) Remark . For a reduced expression w = σ r σ r · · · σ r m in W , we define theelement θ w := θ r θ r · · · θ r m in L . Then it is a well-defined element by the braid relations. Since φ r = θ r P i ∈C q r ( i ) ǫ ( i ),thus by Lemma 3.27 and the decompositions (3.25), it can be proved that L = ⊕ w ∈W , i ∈C k ( y ) θ w ǫ ( i ) = ⊕ w ∈W , i ∈C θ w k ( y ) ǫ ( i ) . (3.38)3.5. The KLR-type generators of L . In the Lusztig extension L , for each r ∈ [ n ],we define an element ψ r = X i ∈C [ θ r − δ i r y − r ] ǫ ( i ) , (3.39)where δ i r = i r = 0 , i r = 0is the Kronecker delta.Using these elements, we can construct a KLR-type generators of L . Theorem 3.40. The algebra L is generated by { y , · · · , y n , ψ , · · · , ψ n , f − , ǫ ( i ) | = f ∈ k [ y ] , i ∈ C} subject to the following relations for all admissible indices: (1) ǫ ( i ) ǫ ( j ) = δ ji ǫ ( i ) , P i ∈C ǫ ( i ) = 1 . HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 11 (2) y r ǫ ( i ) = ǫ ( i ) y r , y r y s = y s y r . (3) For any = f ∈ k [ y ] , there holds that f f − = f − f = 1 . (4) ψ r ǫ ( i ) = ǫ ( σ r ( i )) ψ r . (5) ψ r y s ǫ ( i ) = ( σ r ( y s ) ψ r + δ i r ∂ r ( y s )) ǫ ( i ) . (6) ψ r ǫ ( i ) = if i r = 0; − y r ǫ ( i ) if i r = 1 , e = 2; y r ǫ ( i ) if i r = − , e = 2; − y r ǫ ( i ) if i r = 1 , e = 2; ǫ ( i ) if i r = 0 , ± . (7) If r / − s , then ψ r ψ s = ψ s ψ r . (8) If r − s , then [ ψ r ψ s ψ r − ψ s ψ r ψ s ] ǫ ( i ) = ǫ ( i ) if i r = − i s = 1 , e = 2 , − ǫ ( i ) if i r = − i s = − , e = 2 , ( y r − y s ) ǫ ( i ) if i r = i s = 1 , e = 2 , else . (9) If r i s , then [( ψ r ψ s ) − ( ψ s ψ r ) ] ǫ ( i ) = − ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2; ψ r ǫ ( i ) if i s = − , i r = 2 , e = 2;( y r ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , ψ s ǫ ( i ) if i r = − i s = 1 , e = 2; − ψ s ǫ ( i ) if i r = − i s = − , e = 2; − y s ψ s ǫ ( i ) if i r = i s = 1 , e = 2;0 else . (10) If r i s , then [( ψ r ψ s ) − ( ψ s ψ r ) ] ǫ ( i ) FAN KONG AND ZHI-WEI LI = − ψ r ψ s ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2 , ,ψ r ψ s ψ r ǫ ( i ) if i s = − , i r = 3 , e = 2 , , ψ s ψ r ψ s ǫ ( i ) if i r = − i s = 1 , e = 2 , − ψ s ψ r ψ s ǫ ( i ) if i r = − i s = − , e = 2 , − y r ψ r ǫ ( i ) if i s = ± , i r = ∓ , e = 2 , − y r + 3 y s y s )( y r ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , − y s ψ s ǫ ( i ) if i r = ± , i s = 2 , e = 4 ,ψ s ǫ ( i ) if i r = 1 , i s = − , e = 2 , − ψ s ǫ ( i ) if i r = − , i s = − , e = 2 ,ψ s ǫ ( i ) if i r = 3 , i s = − , e = 2 , , − ψ s ǫ ( i ) if i r = − , i s = 2 , e = 2 , , y r + 3 y s y s )( y s ψ s + 1) ǫ ( i ) if i r = 1 , i s = 0 , e = 2 , ( y s ψ r ψ s ψ r − y r ψ s ψ r ψ s − y r + 3 y s ) ǫ ( i ) if i r = i s = 1 , e = 2 , − ψ r ψ s ψ r ǫ ( i ) if i r = 0 , i s = 1 , e = 3 ,ψ r ψ s ψ r ǫ ( i ) if i r = 0 , i s = − , e = 3 , else . Proof. The statement (1) holds since the definition of the semi-simple algebra E , andthe statements (2)-(3) follow straightforwardly from the construction of the elements y r .(4) By (3.30) and (3.39), we get ψ r ǫ ( i ) = ( θ r − δ i r y − r ) ǫ ( i ) = ǫ ( σ r ( i ))( θ r − δ i r y − r ) = ǫ ( σ r ( i )) ψ r (3.41)where the third identity holds since σ r ( i ) = i whenever i r = 0.(5) As a result of (3.39) and (3.31), it follows that ψ r y s ǫ ( i ) = ( θ r − δ i r y − r ) y s ǫ ( i )= ( σ r ( y s ) θ r − δ i r y s y − r ) ǫ ( i )= ( σ r ( y s ) ψ r + δ i r σ r ( y s ) y − r − δ i r y s y − r ) ǫ ( i )= ( σ r ( y s ) ψ r + δ i r ∂ r ( y s )) ǫ ( i ) . (6) Using (3.39), (3.30) and (3.36), we obtain ψ r ǫ ( i ) = ψ r ǫ ( σ r ( i )) ψ r ǫ ( i ) = ( θ r − δ i r y − r )( θ r − δ i r y − r ) ǫ ( i ) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 13 = ( θ r + δ i r y − r ) ǫ ( i )= i r = 0 , − y r ǫ ( i ) if i r = 1 , e = 2 ,y r ǫ ( i ) if i r = − , e = 2 , − y r ǫ ( i ) if i r = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . (7) Since r / − s , there holds that ψ r ψ s ǫ ( i ) = ψ r ǫ ( σ s ( i )) ψ s ǫ ( i )= ( θ r − δ i r y − r )( θ s − δ i s y − s ) ǫ ( i )= ( θ r θ s − δ i r y − r θ s − δ i s θ r y − s + δ i r δ i s y − r y − s ) ǫ ( i )= ( θ s θ r − δ i r θ s y − r − δ i s y − s θ r + δ i r δ i s y − r y − s ) ǫ ( i )= ( θ s − δ i s y − s )( θ r − δ i r y − r ) ǫ ( i )= ψ s ψ r ǫ ( i )Hence ψ r ψ s = ψ s ψ r .(8) Since r − s , we have ψ r ψ s ψ r ǫ ( i ) = ( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r ) ǫ ( i )= [ θ r θ s θ r − δ i r y − s θ r θ s − δ i s y − r θ s θ r + δ i r δ i s ( y − s y − s θ r + y − r y − r θ s ) − δ i s y − s θ r − δ i r δ i s y − r y − s ] ǫ ( i )Similarly, we can show that ψ s ψ r ψ s ǫ ( i ) = ( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s ) ǫ ( i )= [ θ s θ r θ s − δ i r y − s θ r θ s − δ i s y − r θ s θ r + δ i r δ i s ( y − s y − s θ r + y − r y − r θ s ) − δ i r y − r θ s − δ i r δ i r y − r y − s ] ǫ ( i ) . By (3.33) and (3.36), we arrive at[ ψ r ψ s ψ r − ψ s ψ r ψ s ] ǫ ( i ) = δ i r [ y − r ( θ s − θ r ) + δ i r ( y − r y − s − y − r y − s )] ǫ ( i )= ǫ ( i ) if i r = − i s = 1 , e = 2 , − ǫ ( i ) if i r = − i s = − , e = 2 , ( y r − y s ) ǫ ( i ) if i r = i s = 1 , e = 2 , . FAN KONG AND ZHI-WEI LI (9) Since r i s , by (3.41), (3.39) and (3.31), we deduce that ψ r ψ s ψ r ψ s ǫ ( i )= ( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s ) ǫ ( i )= [( θ r θ s ) − δ i s y − s θ r θ s θ r − δ i r y − r θ s θ r θ s + δ i r δ i s y − r y − s θ r θ s + δ i r δ i s y − r y − s θ s θ r − ( δ i r δ i s y − s y − s + δ i r y − r θ s ) θ r − (2 δ i r δ i s y − r y − r + δ i s y − s θ r ) θ s + δ i r δ i s y − s y − s θ r + δ i r δ i r y − r y − r θ s + δ i r δ i s y − r y − s ] ǫ ( i ) . The following is a similar calculation using (3.41), (3.39) and (3.31): ψ s ψ r ψ s ψ r ǫ ( i )= [( θ s θ r ) − δ i s y − s θ r θ s θ r − δ i r y − r θ s θ r θ s + δ i r δ i s y − r y − s θ r θ s + δ i r δ i s y − r y − s θ s θ r − ( δ i r δ i s y − s y − s + δ i r y − r θ s ) θ r − (2 δ i r δ i s y − r y − r + δ i s y − s θ r ) θ s + δ i r δ i s y − s y − s θ r + δ i r δ i r y − r y − r θ s + δ i r δ i s y − r y − s ] ǫ ( i ) . By (3.34), (3.36) and (3.39), we know that { [ ψ r ψ s ψ r ψ s − ψ s ψ r ψ s ψ r ] ǫ ( i )= [ δ i r δ i s y − s y − s y r + δ i r y − r ( θ s − θ s )] θ r − [ δ i r δ i s y − r y − r y s + δ i s y − s ( θ r − θ r )] θ s } ǫ ( i )= − ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2; ψ r ǫ ( i ) if i s = − , i r = 2 , e = 2;( y r ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , ψ s ǫ ( i ) if i r = − i s = 1 , e = 2; − ψ s ǫ ( i ) if i r = − i s = − , e = 2; − y s ψ s ǫ ( i ) if i r = i s = 1 , e = 2;0 else . (10) Since r i s , as a result of (3.41), (3.39) and (3.31), the following holds:( ψ r ψ s ) ǫ ( i )= ( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s ) ǫ ( i )= { ( θ r θ s ) − δ i s y − s ( θ r θ s ) θ r − δ i r y − r ( θ s θ r ) θ s + δ i r δ i s y − r y − s ( θ s θ r ) + δ i r δ i s y − r y − s ( θ r θ s ) − ( δ i r δ i s y − s y − s + δ i r y − r θ s ) θ r θ s θ r − ( δ i r δ i s y − r y − r + δ i s y − s θ r )] θ s θ r θ s + δ i r [ δ i s (3 y − r y − r y − s + y − r y − s + 2 y − r y − r y − s − y − s y − s θ r + y − s y − s σ r ( θ r ))+ δ i r y − r ( y − r θ s − y − r σ r σ s ( θ s ))] θ r θ s + δ i r [ δ i s (2 y − r y − s + y − s y − s θ r ) + δ i r y − r y − r θ s ] θ s θ r HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 15 + [ δ i r δ i s y − r (2 y − r y − s + y − r y − s − y − r y − s ) θ s + δ i r δ i s y − r y − s ( y − s y − s − y − r y − s − y − r θ s − y r y − s y − s σ r ( θ r )) − δ i s y − s θ r θ s ] θ r + [ δ i r δ i s ( y − r y − s y − s θ r − y − r y − r y − s θ r − y − s y − s θ r )+ δ i r δ i s y − r y − s (3 y − r − y − r y − s ) − δ i r y − r θ s θ r − δ i r δ i r δ i r y − r y − r σ s ( θ s ))] θ s + δ i r δ i s (2 y − r y − r y − s θ s − y − r y − r y − s θ s + y − s y − s θ r + y − r y − s y − s θ r + y − r y − s )+ δ i r δ i s y − r y − s θ s θ r + δ i r δ i s y − r y − s θ r θ s + δ i r δ i s y − r y − s θ r θ s } ǫ ( i )Imitate the proof above, we have that( ψ s ψ r ) ǫ ( i )= ( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r )( θ s − δ i s y − s )( θ r − δ i r y − r ) ǫ ( i )= { [( θ s θ r ) − δ i s y − s ( θ r θ s ) θ r − δ i r y − r ( θ s θ r ) θ s + δ i r δ i s y − r y − s ( θ r θ s ) + δ i r δ i s y − r y − s ( θ s θ r ) − ( δ i r δ i s y − s y − s + δ i r y − r θ s ) θ r θ s θ r − ( δ i r δ i s y − r y − r + δ i s y − s θ r ) θ s θ r θ s + δ i r [ δ i s (2 y − r y − s + y − s y − s θ r ) + δ i r y − r y − r θ s ] θ r θ s + δ i r [ δ i s y − r ( y − r y − s + y − s y − s ) + δ i s y − s (2 y − r y − s + y − s θ r − y − s σ s σ r ( θ r )) + δ i r y − r ( y − r σ s ( θ s ) − y − r θ s )] θ s θ r + [ δ i r δ i s ( y − r y − r y − s θ s − y − r y − s θ s − y − r y − r y − s θ s + y − r y − s y − s − y − r y − s − y − s y − s σ r ( θ r )) − δ i s y − s θ r θ s ] θ r + [ δ i r δ i s (2 y − r y − s y − s + y − r y − s − y − r y − s y − s ) θ r − δ i r y − r θ s θ r − δ i r δ i r δ i r y − r y − r σ s ( θ s ) + δ i r δ i s (3 y − r y − r y − s − y − r y − s y − s θ r − y − r y − r y − s )] θ s + δ i r δ i s [(2 y − r y − s y − s − y − r y − s y − s ) θ r + (3 y − r y − r + y − r y − r y − s ) θ s + y − r y − s ] + ( δ i r δ i s y − r y − s θ r + δ i r δ i s y − r y − s θ r ) θ s + δ i r δ i s y − r y − s θ s θ r } ǫ ( i )Using relations (3.35), (3.36) and (3.39), we see that[( ψ r ψ s ) − ( ψ s ψ r ) ] ǫ ( i )= { [ δ i r δ i s y − s y − s y s + δ i r y − r ( θ s − θ s )] θ r θ s θ r − [ δ i r δ i s y − r y − r y r + δ i s y − s ( θ r − θ r )] θ s θ r θ s + [ δ i r δ i s (3 y − r y − r y − s + y − r y − s + 2 y − r y − r y − s y r ) + δ i r δ i s y − s y − s ( σ r ( θ r ) − θ r )+ δ i r δ i r y − r y − r ( θ s − σ r σ s ( θ s ))] θ r θ s − [ δ i r δ i s (2 y − r y − s y − s y s + y − r y − s + y − r y − s y − s )+ δ i r δ i r y − r y − r ( σ s ( θ s ) − θ s ) + δ i r δ i s y − s y − s ( θ r − σ s σ r ( θ r ))] θ s θ r + [ δ i r δ i s ( y − r y − s y − s θ s + 2 y − r y − s θ s + y − r y − s θ s − y − r y − s y − s θ s + y − s y − s y r σ r ( θ r ) + y − s y − s + y − r y − s y − s )+ δ i s y − s ( θ r θ s − θ r θ s )] θ r + [ δ i r δ i s (6 y − r y − r y − s θ r − y − r y − s θ r + 9 y − r y − r − y − r y − s θ r − y − r y − r y − s θ r − y − r y − r y − s ) − δ i r δ i r y − r y − r y s σ s ( θ s ) + δ i r y − r ( θ r θ s − θ r θ s )] θ s } ǫ ( i ) FAN KONG AND ZHI-WEI LI = δ i r θ s − θ s y r θ r θ s θ r ǫ ( i ) if i r = 0 , i r = 0 , i s = 0 ,δ i s θ r − θ r y s θ s θ r θ s ǫ ( i ) if i s = 0 , i r = 0 , i r = 0 ,δ i s θ r θ s − θ r θ s y s θ r ǫ ( i ) if i s = 0 , i r = 0 ,δ i r θ r θ s − θ r θ s y r θ s ǫ ( i ) if i r = 0 , i r = 0 ,δ i s θ r θ s − θ r θ s y s θ r ǫ ( i ) if i s = 0 , i r = 0 , i s = 0 ,δ i r θ r θ s − θ r θ s y r θ s ǫ ( i ) if i r = 0 , i s = 0 , i r = 0 , ( δ i r θ s − θ s y r θ r θ s θ r + δ i s θ r − θ r y s θ s θ r θ s ) ǫ ( i ) if i r = 0 , i s = 0 , i r = 0 , ( δ i r θ s − θ s y r θ r θ s θ r + δ i r δ i r σ r σ s ( θ s ) − θ s σ r ( y r ) y r θ r θ s + δ i r δ i r θ s − σ s ( θ s ) y r y r θ s θ r − δ i r δ i r y s σ s ( θ s ) y r y r θ s + δ i r θ r θ s − θ r θ s y r θ s ) ǫ ( i ) if i r = 0 , i r = 0 , i r = 0 , i s = 0 , − ψ r ψ s ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2 , ,ψ r ψ s ψ r ǫ ( i ) if i s = − , i r = 3 , e = 2 , , ψ s ψ r ψ s ǫ ( i ) if i r = − i s = 1 , e = 2 , − ψ s ψ r ψ s ǫ ( i ) if i r = − i s = − , e = 2 , − y r ψ r ǫ ( i ) if i s = ± , i r = ∓ , e = 2 , − y s ψ s ǫ ( i ) if i r = ± , i s = 2 , e = 4 ,ψ s ǫ ( i ) if i r = 1 , i s = − , e = 2 , − ψ s ǫ ( i ) if i r = − , i s = − , e = 2 ,ψ s ǫ ( i ) if i r = 3 , i s = − , e = 2 , , − ψ s ǫ ( i ) if i r = − , i s = 2 , e = 2 , , − y r + 3 y s y s )( y r ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , y r + 3 y s y s )( y s ψ s + 1) ǫ ( i ) if i r = 1 , i s = 0 , e = 2 , ( y s ψ r ψ s ψ r − y r ψ s ψ r ψ s − y r + 3 y s ) ǫ ( i ) if i r = i s = 1 , e = 2 , − ψ r ψ s ψ r ǫ ( i ) if i r = 0 , i s = 1 , e = 3 ,ψ r ψ s ψ r ǫ ( i ) if i r = 0 , i s = − , e = 3 , . To finish the proof of the theorem, we need to prove the relations (1)-(10) gen-erate all relations. In fact, for each w ∈ W , we fix a reduced decomposition HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 17 w = σ r σ r · · · σ r m and define the element ψ w := ψ r ψ r · · · ψ r m ∈ L . Note that ψ w in general does depend on the choice of reduced decomposition of w [4, Proposition 2.5]. Using the decomposition (3.38), L has a k ( y ) ⊗ k E -basis { θ w | w ∈ W} . By (3.39), we know that { ψ w | w ∈ W} is also a basis of L as k ( y ) ⊗ k E -module. Thus the relations (1)-(9) is complete since by them every elementin L can be written as P w ∈W , i ∈C ψ w f w, i ( y ) g − w, i ( y ) ǫ ( i ) with f w, i ( y ) , g w, i ( y ) ∈ k [ y ] and g w, i ( y ) = 0. (cid:3) The BK subalgebras. Our method of constructing the KLR generators ofthe Lusztig extension follows Brundan and Kleshchev. With these KLR generatorsin hand, we can fast obtain a class of BK-type isomorhism. In the process, the keypoint is the following subalgebra of the Lusztig extension using its KLR form. Wethink it is suitable to call it Brundan Kleshchev subalgebra (or BK subalgebra forshort).We define the BK subalgebra ˜ L as the k -algebra generated by { y , · · · , y n , ψ , · · · , ψ n , f − ( y ) , ǫ ( i ) | i ∈ C , f ( y ) ∈ k [ y ] with f (0) = 0 } subject to the relations (1)-(10) of Theorem 3.40. It is a subalgebra of L as shownin the following corollary. Corollary 3.42. Denote by ǫ ( i ) = ǫ ( i ) . Then the algebra ˜ L is generated by { x , · · · , x n , t , · · · , t n , f − ( x ) ǫ ( i ) | i ∈ C , f ( x ) ∈ k [ x ] with f ( i ) = 0 } subject to relations (3.1)-(3.7), (3.20), (3.21), (3.23) and for f ∈ k [ X ] with f ( i ) = 0 ǫ ( j ) · f − ǫ ( i ) = δ ji f − ǫ ( i ) = f − ǫ ( i ) · ǫ ( j ) , f · f − ǫ ( i ) = ǫ ( i ) = f − ǫ ( i ) · f. (3.43) Proof. There is an obvious homomorphism α : ˜ L → L by sending generators tothe same named generators. This homomorphism is injective since using relations(1)-(9) of Theorem 3.40, every element in ˜ L can be written as X w ∈W , i ∈C ψ w f w, i ( y ) · g − w, i ( y ) ǫ ( i )with f w, i ( y ) , g w, i ( y ) in k [ y ] and g w, i (0) = 0, and { ψ w | w ∈ W} is a k ( y ) ⊗ k E -basis of L . Thus Im α = ⊕ w ∈W ψ w P ( y, E ), where P ( y, E ) is the commutative algebra { f g − | f, g ∈ k [ y ] , g (0) = 0 } ⊗ k E . FAN KONG AND ZHI-WEI LI Assume M is the k -algebra generated by { x , · · · , x n , t , · · · , t n , f − ( x ) ǫ ( i ) | i ∈ C , f ( x ) ∈ k [ x ] with f ( i ) = 0 } subject to relations (3.1)-(3.7), (3.20), (3.21), (3.23) and (3.43). Then, using theserelations, every element in M can be written as X w ∈W , i ∈C T w f w, i ( x ) · g − w, i ( x ) ǫ ( i )with f w, i ( x ) , g w, i ( x ) in k [ x ] and g w, i ( i ) = 0. Thus there also has an injectivehomomorphism α ′ : M → L by sending generators to the same named gener-ators. So Im α ′ = ⊕ w ∈W T w P ( x, E ), where P ( x, E ) is the commutative algebra { f · g − ǫ ( i ) | i ∈ C , f, g ∈ k [ x ] , g ( i ) = 0 } .We claim that Im α = Im α ′ . In fact, by Lemma 3.27, P ( x, E ) = P ( y, E ) in L .Notice that ψ r = X i ∈C i r =0 ( t r + y r + i r ) q − r ( i ) ǫ ( i ) + X i ∈C i r =0 ( t r + 1) q − r ( i ) ǫ ( i ) . and q r ( i ) , q − r ( i ) ∈ P ( y, E ) in L , thus we get Im α = Im α ′ and then M ∼ = ˜ L . (cid:3) Denote by ˜ L (Λ) := ˜ L / h y Λ i ǫ ( i ) | i ∈ Ci . We use the same letters ψ , · · · , ψ n and y , · · · , y n to denote the images of thegenerators in ˜ L (Λ). Lemma 3.44. ˜ L (Λ) = ˜ L / h Q i ∈ I ( x − i ) Λ i i . Proof. By the relations of y and x , there holds that Y i ∈ I ( x − i ) Λ i = X j ∈C Y i ∈ I ( y + j − i ) Λ i ǫ ( j )= X j ∈C Y i ∈ I,i = j ( y + j − i ) Λ i y Λ j ǫ ( j )is in h y Λ j ǫ ( j ) | j ∈ Ci . By Corollary (3.42), Q i ∈ I,i = j [( x − i ) Λ i ] − ǫ ( j ) is in ˜ L , thus y Λ j ǫ ( j ) = ( x − j ) Λ j ǫ ( j )= Y i ∈ I ( x − i ) Λ i Y i ∈ I,i = j [( x − i ) Λ i ] − ǫ ( j )is in h Q i ∈ I ( x − i ) Λ i i . Therefore h y Λ i ǫ ( i ) | i ∈ Ci = h Q i ∈ I ( x − i ) Λ i i in ˜ L . Thus˜ L (Λ) = ˜ L / h Q i ∈ I ( x − i ) Λ i i . (cid:3) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 19 Remark . Similarly to the proof of [3, Lemma 2.1], one can deduce that theelements y r are nilpotent in ˜ L (Λ). Then by imitating the proof above, we know thatthe elements Q i ∈ I ( x r − i ) are also nilpotent in ˜ L (Λ).3.7. Cyclotomic degenerate affine Hecke algebras. We define the cyclotomicdegenerate affine Hecke algebra H (Λ) as H (Λ) := H / h Y i ∈ I ( x − i ) Λ i i . Similar to [3, Subsection 3.1], there is a system { e ( i ) | i ∈ C} of mutually orthogonalidempotents in H (Λ) such that 1 = P i ∈ I n e ( i ) and e ( i ) H (Λ) = { h ∈ H (Λ) | ( x r − i r ) m h = 0 for all r ∈ [ n ] and m ≫ } . It is easy to check that x r e ( i ) = e ( i ) x r for all r ∈ [ n ] and i ∈ C , and for a polynomial f ( x ) ∈ k [ x ], f ( x ) e ( i ) is a unit if and only if f ( i ) = 0. In particular, the element x r e ( i ) is a unit in e ( i ) H (Λ) if and only if i r = 0. In this case, we write x − r e ( i ) forthe inverse. Lemma 3.46. The following relations hold for all r ∈ [ n ] and i ∈ I n . t r e ( i ) = e ( i ) t r if i r = 0 ,e ( σ r ( i )) t r + x − r e ( σ r ( i )) − x − r e ( i ) if i r = 0 . (3.47) Proof. For any s ∈ [ n ], the element( σ r ( x s ) − σ ( i ) s ) e ( i ) = [( x s − i s ) − a sr ( x r − i r )] e ( i )is nilpotent by the nilpotency of ( x s − i s ) e ( i ) and ( x r − i r ) e ( i ). Similarly, we canshow that ∂ r (( x s − i s ) m ) e ( i ) is nilpotent by the binomial theorem for an integer m ≫ i r = 0. Therefore, if i r = 0, by (3.8), we have( x s − i s ) m t r e ( i ) = t r ( σ r ( x s ) − i s ) m e ( i ) + ∂ r (( x s − i s ) m ) e ( i ) = 0when m ≫ 0. Hence t r e ( i ) ∈ e ( i ) H (Λ) and then t r e ( i ) = e ( i ) t r e ( i ) = e ( i ) t r . If i r = 0, by (3.8), we get that( x s − σ r ( i ) s ) m [ t r x r + 1] e ( i ) = [ t r x r + 1]( σ r ( x s ) − σ r ( i ) s ) m e ( i ) = 0when m ≫ 0. Therefore t r x r e ( i ) + e ( i ) = e ( σ r ( i ))[ t r x r e ( i ) + e ( i )] = e ( σ ( i )) t r x r e ( i ) . FAN KONG AND ZHI-WEI LI Then right-multiplying by x − r e ( i ), we obtain that t r e ( i ) = e ( σ r ( i )) t r e ( i ) − x − r e ( i ) . Similarly, we deduce that e ( σ r ( i )) t r = e ( σ r ( i )) t r e ( i ) − x − r e ( σ r ( i )) . Therefore t r e ( i ) = e ( σ r ( i )) t r + x − r e ( σ r ( i )) − x − r e ( i ) . (cid:3) Let e ( C ) := X i ∈C e ( i ) ∈ H (Λ) . Then e ( C ) is a central idempotent in H (Λ) by (3.47). Furthermore, by Lemma 3.44,Corollary 3.42 and Lemma 3.46, there is a homomorphism ρ : ˜ L (Λ) → H (Λ) e ( C )sending the generators x r , t r to the same named elements, and f ( x ) − ǫ ( i ) with f ( i ) =0 to f − ( x ) e ( i ). Now we arrive at our first main result in this article for degenerateaffine Hecke algebras. Theorem 3.48. There is an algebra isomorphism H (Λ) e ( C ) ∼ = ˜ L (Λ) .Proof. Apparently, ρ is surjective, thus we only need to construct a left-inverse of ρ .By lemma 3.44, there is a homomorphism τ : H (Λ) → ˜ L (Λ)sending the generators x r , t r to the same named elements. Let i ∈ C and j ∈ I n . If i = j , then there is some 1 ≤ r ≤ n such that j r = i r . We claim that ǫ ( i ) τ ( e ( j )) = 0. In fact, by the construction of e ( j ), there is an integer m ≫ x r − j r ) m e ( j ) = 0. So we have( x r − j r ) m ǫ ( i ) τ ( e ( j )) = ǫ ( i ) τ (( x r − j r ) m e ( j )) = 0 . The assumption j r = i r implies that the element ( x r − j r ) − ǫ ( i ) ∈ ˜ L (Λ). Thus wededuce that ǫ ( i ) τ ( e ( j )) = ( x r − j r ) − m ( x r − j r ) m ǫ ( i ) τ ( e ( j )) = ( x r − j r ) − m ǫ ( i )0 = 0 . Therefore, if j ∈ I n \ C we have that τ ( e ( j )) = P i ∈C ǫ ( i ) τ ( e ( j )) = 0 . Moreover, if j ∈ C , we get that τ ( e ( j )) = X i ∈C ǫ ( i ) τ ( e ( j )) = ǫ ( j ) τ ( e ( j )) = ǫ ( j ) X i ∈ I n τ ( e ( i )) = ǫ ( j ) τ (1) = ǫ ( j ) . HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 21 These show that τ | H (Λ) e ( C ) : H (Λ) e ( C ) → ˜ L (Λ) is an algebra homomorphism. Itis easy to check that τ ρ is the identity on each generator of ˜ L (Λ). Thus ρ is anisomorphism. (cid:3) The cyclotomic BK subalgbra revisited. Following [3, Subsection 2.2], weintroduce an algebra R which is defined to be the k -algebra R generated by { y , · · · , y n , ψ , · · · , ψ n , ǫ ( i ) | i ∈ C , } subject to the relations (1)-(2) and (4)-(10) of Theorem 3.40. Similar to the Lusztigextension L , by the defining relations, R has a basis { ψ w | w ∈ W} as a k [ y ] ⊗ k E -module. Moreover, it can be viewed as a subalgebra of L .We denote by R (Λ) := R / h y Λ i ǫ ( i ) | i ∈ Ci . Then we have the following isomor-phism of cyclotomic algebras. Proposition 3.49. We have k -algebra isomorphsim R (Λ) ∼ = ˜ L (Λ) .Proof. Similarly to the proof of [3, Lemma 2.1], we can also show that the elements y r are nilpotent in R (Λ). Thus if f ( y ) ∈ k [ y ] with f (0) = 0, the polynomial f ( y ) − f (0)is nilpotent in ˜ L (Λ) by the nilpotency of the elements y r . So there exists some g ( y ) ∈ k [ y ] and m ∈ N such that g ( y ) m = 0 and f − ( y ) = f (0) − P ml =0 g ( y ) l in R (Λ).Therefore, the homomorphism R ֒ → ˜ L ։ ˜ L (Λ) is surjective and induces a surjectivehomomorphism π : R (Λ) → ˜ L (Λ). Let ˜ F be the localization of the commutative ring k [ y ] W of W -invariants in k [ y ] with respect to { f ∈ k [ y ] W | f (0) = 0 } . Similar to theproof of (2.2), we can show that ˜ L ∼ = R ⊗ k [ y ] W ˜ F . Since the elements y r are nilpotentin R (Λ), similar to the proof above, we know that if f ( y ) ∈ K [ y ] W with f (0) = 0,then it is a unit in R (Λ). Thus the homomorphism k [ y ] W ֒ → R π ։ R (Λ) inducesa morphism π : ˜ F → R (Λ). Therefore we have an induced algebra homomorphism π ⊗ π : ˜ L → R (Λ). The homomorphism π ⊗ π induces an algebra homomorphism π ′ : ˜ L (Λ) → R (Λ). It is easy to check that π and π ′ are two-sided inverses. So R (Λ) ∼ = ˜ L (Λ). (cid:3) By Theorem 3.48 and the Proposition above, we get a Brundan-Kleshchev typeisomorphism for the degenerate affine Hecke algebra. Corollary 3.50. We have k -algebra isomorphism H (Λ) e ( C ) ∼ = R (Λ) . The BK subalgebra in non-degenerate case In this section, we assume that e is the smallest positive integer such that 1 + q + · · · + q e − = 0 and setting e = 0 if no such that integer exists. FAN KONG AND ZHI-WEI LI The non-degenerate affine Hecke algebras. We define the non-degenerateaffine Hecke algebra H q to be the unital k -algebra with generators { X ± r , T r | r ∈ [ n ] } subject to the following relations for all admissible indices: X r X s = X s X r , X r X − r = X − r X r = 1; (4.1) T r X s = σ r ( X s ) T r + (1 − q )D r ( X s ); (4.2)( T r + 1)( T r − q ) = 0; (4.3) T r T s = T s T r , if r / − s ; (4.4) T r T s T r = T s T r T s , if r − s ; (4.5)( T r T s ) = ( T s T r ) , if r i s ; (4.6)( T r T s ) = ( T s T r ) , if r i s . (4.7)If w = σ r σ r · · · σ r m is a reduced expression in W , then T w := t r t r · · · t r m is a well-defined element in H q . By [7, Lemma 3.4], the algebra H q has Bernstein-Zelevinski basis { T w | w ∈ W} as k [ X ± ]-module.4.2. The rationalization of H q . Recall [7, proposition 3.11] that the center of H q is Z . Then H can be seen as a Z -subalgebra (identified with the subspace H q ⊗ Z -algebra H q, F := H q ⊗ Z F . Moreover, we can identify the rational polynomial field k ( X ) as a subspace of H q, F via the natural isomorphism (2.5). For any r ∈ [ n ] and f ∈ k ( X ), as a consequenceof [7, Subsection 3.12 (d)], there holds that T r f = σ r ( f ) T r + (1 − q )D r ( f ) . (4.8)4.3. Intertwining elements. For r ∈ [ n ], we define the intertwining element Φ r in H q, F as follows: Φ r := T r + (1 − q )(1 − X r ) − . The following result is the non-degenerate version of Proposition 3.10. Its proof issimilarly to the degenerate case. Proposition 4.9. The algebra H q, F is generated by { X , · · · , X n , Φ , · · · , Φ n , f − | = f ∈ k [ X ] } subject to the following relations for all admissible r, s : X r X s = X s X r ; (4.10) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 23 f f − = f − f = 1 ∀ = f ∈ k [ X ]; (4.11)Φ r X s = σ r ( X s )Φ r ; (4.12)Φ r = (1 − qX r )( q − X r )(1 − X r ) ; (4.13)Φ r Φ s = Φ s Φ r if r / − s ; (4.14)Φ r Φ s Φ r = Φ s Φ r Φ s if r − s ; (4.15)(Φ r Φ s ) = (Φ s Φ r ) if r i s ; (4.16)(Φ r Φ s ) = (Φ s Φ r ) if r i s . (4.17)If w = σ r σ r · · · σ r m is a reduced expression in W , then we have a well-definedelement Φ w := Φ r Φ r · · · Φ r m ∈ H q, F . The Proposition above shows that H q,F has decompositions H q,F = ⊕ w ∈W k ( X ) T w = ⊕ w ∈W T w k ( X ) (4.18)and H q,F = ⊕ w ∈W k ( X )Φ w = ⊕ w ∈W Φ w k ( X ) . (4.19)4.4. The Lusztig extension of H q . Let E be the unital k -algebra as defined in(3.20). The Lusztig extension of H q with respect to E is the k -algebra L q which isequal as k -space to the tensor product L := H q,F ⊗ k E = ⊕ w ∈W , i ∈C Φ w k ( X ) ǫ ( i )of the rationalization algebra H q,F and the semi-simple algebra E . Multiplication isdefined so that H q,F (identified with the subspace H q,F ⊗ 1) and E (identified withthe subspace 1 ⊗ E ) are subalgebras of L q , and in addition X r ǫ ( i ) = ǫ ( i ) X r , (4.20)Φ r ǫ ( i ) = ǫ ( σ r ( i ))Φ r . (4.21)For r ∈ [ n ] and i ∈ C , it is easy to see that T r ǫ ( i ) = ǫ ( σ r ( i )) T r + (1 − q )(1 − X r ) − ǫ ( σ r ( i )) − (1 − q )(1 − X r ) − ǫ ( i ) . (4.22)Similar to the degenerate case, the algebra L q has decompositions L q = ⊕ w ∈W , i ∈C k ( X ) T w ǫ ( i ) = ⊕ w ∈W , i ∈C T w k ( X ) ǫ ( i ) (4.23)and L q = ⊕ w ∈W , i ∈C k ( X )Φ w ǫ ( i ) = ⊕ w ∈W , i ∈C Φ w k ( X ) ǫ ( i ) . (4.24) FAN KONG AND ZHI-WEI LI Brundan-Kleshchev auxiliary elements. Recall that [3, (4.21)], for each r ∈ [ n ], Brundan and Kleshchev introduced the elements Y r := X i ∈C (1 − q − i r X r ) ǫ ( i ) (4.25)Then Y r ǫ ( i ) = ǫ ( i ) Y r by 4.20, and Y r is a unit in L q with Y − r = X i ∈C (1 − q − i r X r ) − ǫ ( i ) . Let k [ Y ± ] be the Laurent polynomial ring with Y := Y , Y , · · · , Y n and k ( Y )be the rational function field. Similar to the proof of Lemma 3.27, we know that k ( X ) ⊗ k E = k ( Y ) ⊗ k E in L q .We observe that there is an action of W on k [ Y ± ] (by the ring automorphism)since σ r ( Y s ) = X i ∈C ( σ r ( X s ) − q i s ) ǫ ( σ r ( i )) = 1 − (1 − Y s )(1 − Y r ) − a sr for all r, s ∈ [ n ]. It can be extended to an action of W on the rational function field k ( Y ) via the field automorphism w ( f ( Y ) g ( Y ) ) = f ( w ( Y )) g ( w ( Y )) for any w ∈ W . For simplicity,for each r ∈ [ n ], we shall write Y r = σ s ( Y r ) , Y r = σ r σ s ( Y r ) , Y r = σ s σ r σ s ( Y r ) , Y r = σ r σ s σ r σ s ( Y r ) , Y r = σ s σ r σ s σ r σ s ( Y r ) in the sequel.Following Brundan and Kleshchev, for each r ∈ [ n ], we define the element Q r ( i ) = − q − Y r if i r = 0 , i r = 1 , e = 2 , [ q − q − (1 − Y r )][1 − q − (1 − Y r )] − if i r = − , e = 2 , [ q (1 − Y r ) − − if i r = 1 , e = 2 , [ q i r (1 − Y r ) − q ][ q i r (1 − Y r ) − − if i r = 0 , ± . (4.26)Similar to the proof of [3, (4.33)-(4.35)], we can deduce the following result. Lemma 4.27. Let r, s ∈ [ n ] and i ∈ C . Then σ r ( Q r ( σ r ( i ))) Q r ( i ) = [(1 − Y r ) − q ][1 − q (1 − Y r )]1 − y r if i r = 0; q ( Y r − − q (1 − Y r )][1 − q (1 − Y r )] if i r = 1 , e = 2; q − q − (1 − Y r )[1 − q − (1 − Y r )] if i r = − , e = 2; − q (1 − Y r )[1 − q (1 − Y r )] if i r = 1 , e = 2; [ q − q ir (1 − Y r )][1 − q ir (1 − Y r )][1 − q ir (1 − Y r )] if i r = 0 , ± . (4.28) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 25 Inside L q , set Θ r = Φ r X i ∈C Q − r ( i ) ǫ ( i ) . These elements have the following nice properties by using relations (4.10)-(4.17):Θ r ǫ ( i ) = ǫ ( σ r ( i ))Θ r ; (4.29) Y s Θ r = Θ r σ r ( Y s ); (4.30)Θ r Θ s = Θ s Θ r if r / − s ; (4.31)Θ r Θ s Θ r = Θ s Θ r Θ s if r − s ; (4.32)(Θ r Θ s ) = (Θ s Θ r ) if r i s ; (4.33)(Θ r Θ s ) = (Θ s Θ r ) if r i s (4.34)and Θ r ǫ ( i ) = ( Y r − Y − r ǫ ( i ) if i r = 0 ,Y r ( Y r − − ǫ ( i ) if i r = 1 , e = 2 ,Y r ǫ ( i ) if i r = − , e = 2 ,Y r ( Y r − − ǫ ( i ) if i r = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . (4.35)In L q , we have a well-defined element Θ w := Θ r Θ r · · · Θ r m if w = σ r σ r · · · σ r m is a reduced expression in W . Since Φ r = Θ r P i ∈C Q r ( i ) ǫ ( i ), thus by the decompo-sitions (4.24) and k ( X ) ⊗ k E = k ( Y ) ⊗ k E , we obtain that L q = ⊕ w ∈W , i ∈C k ( Y )Θ w ǫ ( i ) = ⊕ w ∈W , i ∈C Θ w k ( Y ) ǫ ( i ) . (4.36)4.6. The KLR-type generators of L q . In L q , for each r ∈ [ n ], setΨ r = X i ∈C [Θ r − δ i r Y − r ] ǫ ( i ) . (4.37)Using these elements, we can construct a KLR-type generators of L q . Theorem 4.38. The algebra L q is generated by { Y , · · · , Y n , Ψ , · · · , Ψ n , f − , ǫ ( i ) | = f ∈ k [ Y ] , i ∈ C} subject to the following relations for all admissible indices: (1) ǫ ( i ) ǫ ( j ) = δ ji ǫ ( i ) , P i ∈C ǫ ( i ) = 1 . (2) Y r ǫ ( i ) = ǫ ( i ) Y r , Y r Y s = Y s Y r . (3) For any = f ∈ k [ Y ] , there holds that f f − = f − f = 1 . (4) Ψ r ǫ ( i ) = ǫ ( σ r ( i ))Ψ r . (5) Ψ r Y s ǫ ( i ) = [ σ r ( Y s ) ψ r + δ i r ∂ r ( Y s )] ǫ ( i ) . FAN KONG AND ZHI-WEI LI (6) Ψ r ǫ ( i ) = − Ψ r ǫ ( i ) if i r = 0 ,Y r ( Y r − − ǫ ( i ) if i r = 1 , e = 2 ,Y r ǫ ( i ) if i r = − , e = 2 ,Y r ( Y r − − ǫ ( i ) if i r = 1 , e = 2 ,ǫ ( i ) if i r = 0 , ± . (7) If r / − s , then Ψ r Ψ s = Ψ s Ψ r . (8) If r − s , then (Ψ r Ψ s Ψ r − Ψ s Ψ r Ψ s ) ǫ ( i ) = − Y r ǫ ( i ) if i r = − i s = 1 , e = 2 , Y s − ǫ ( i ) if i r = − i s = − , e = 2 , Y r − Y s (1 − Y r )(1 − Y s ) ǫ ( i ) if i r = i s = 1 , e = 2 , else . (9) If r i s , then [(Ψ r Ψ s ) − (Ψ s Ψ r ) ] ǫ ( i ) = Y s − Ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2; − Y s Ψ r ǫ ( i ) if i s = − , i r = 2 , e = 2; − Y s ( Y r Ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , − Y s − Y r Ψ s ǫ ( i ) if i r = − i s = 1 , e = 2; − Y s Y r − Ψ s ǫ ( i ) if i r = − i s = − , e = 2; Y s (2 − Y s )( Y s − − Y r Ψ s ǫ ( i ) if i r = i s = 1 , e = 2;0 else . (10) If r i s , then [(Ψ r Ψ s ) − (Ψ s Ψ r ) ] ǫ ( i ) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 27 = Y s − Ψ r Ψ s Ψ r ǫ ( i ) if i r = − , i s = 1 , e = 2 , , − Y s Ψ r Ψ s Ψ r ǫ ( i ) if i r = 3 , i s = − , e = 2 , , − Y s + Y s − Y r Ψ s Ψ r Ψ s ǫ ( i ) if i r = − i s = 1 , e = 2 , − Y s + Y s Y r − Ψ s Ψ r Ψ s e ( i ) ǫ ( i ) if i r = − i s = − , e = 2 , (2 − Y s )( Y s − Y s + Y r Y s )(1 − Y r )(1 − Y s ) Ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2 , (2 − Y s )( Y r − Y r + Y s Y r )(1 − Y r )(1 − Y s ) Ψ r ǫ ( i ) if i s = − , i r = 2 , e = 2 , Y s ( Y s − − Y s Ψ s ǫ ( i ) if i r = 1 , i s = 2 , e = 4 , Y s ( Y s − − Y r Ψ s ǫ ( i ) if i r = − , i s = 2 , e = 4 , Y s − Ψ s ǫ ( i ) if i r = 3 , i s = − , e = 2 , , − Y s Ψ s ǫ ( i ) if i r = − , i s = 2 , e = 2 , , Y r − Ψ s ǫ ( i ) if i r = 1 , i s = − , e = 2 , − Y r Ψ s ǫ ( i ) if i r = − , i s = − , e = 2 , ( Y s − Y r Y s + Y r Y s (1 − Y r )](1 − Y s )(1 − Y r ) ( Y r ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , (2 − Y s )[ Y r Y s + Y r Y s (1 − Y s ) ](1 − Y r )(1 − Y s ) ( Y s ψ s + 1) ǫ ( i ) if i r = 1 , i s = 0 , e = 2 , [ Y s − Y s Ψ r Ψ s Ψ r − Y r (3 − Y s + Y s )1 − Y r Ψ s Ψ r Ψ s − Y r (1 − Y s ) + Y s (3 − Y s + Y s )(1 − Y s ) ] ǫ ( i ) if i r = i s = 1 , e = 2 , Y s − Ψ r Ψ s Ψ r ǫ ( i ) if i r = 0 , i s = 1 , e = 3 , − Y s (Ψ r Ψ s Ψ r + Ψ r Ψ s ) ǫ ( i ) if i r = 0 , i s = − , e = 3 , else . Proof. The proofs of Statements (1)-(9) are similarly to the corresponding State-ments of Theorem 3.40 in the degenerate case, hence we skip them. Here we onlygive the proof of the Statement (10) which is the most complicated one.(10) Since r i s , by Statement (3), relations (4.37) and (4.30), we get that(Ψ r Ψ s ) ǫ ( i )= (Θ r − δ ir Y r )(Θ s − δ is Y s )(Θ r − δ ir Y r )(Θ s − δ is Y s )(Θ r − δ ir Y r )(Θ s − δ is Y s ) ǫ ( i )= { (Θ r Θ s ) − δ is Y s (Θ r Θ s ) Θ r − δ ir Y r Θ s (Θ r Θ s ) + δ ir δ is Y r Y s (Θ r Θ s ) + δ ir δ is Y r Y s (Θ s Θ r ) − ( δ ir δ is Y s Y s + δ ir Θ s Y r )Θ r Θ s Θ r − ( δ is Θ r Y s + δ ir δ is (3 − Y s + Y s ) Y r Y r )Θ s Θ r Θ s + δ i r [ δ i s ( Θ r σ r σ s ( Y s ) Y s + σ r (Θ r ) Y s Y s + ( Y s − Y r Y s + − Y s − Y s σ r ( Y r ) Y r Y s + − Y s Y r Y r Y s ) + δ i r ( σ r σ s (Θ s ) σ r ( Y r ) Y r + Θ s Y r Y r )]Θ r Θ s + δ i r [ δ i s ( Θ r Y s Y s + − Y s Y r Y s ) FAN KONG AND ZHI-WEI LI + δ ir Θ s Y r Y r )]Θ s Θ r − [ δ i r δ i s ( (3 − Y s − Y s )Θ s σ r ( Y r ) Y r + Θ s Y r Y s Y s + σ r (Θ r ) Y s Y s + Θ s Y r Y r Y s + σ r ( Y r ) Y s + Y r Y s Y s )+ δ is Θ r Θ s Y s ]Θ r − [ δ i r δ i s ( (2 − Y s )Θ r Y r Y s + Θ r Y r Y s Y s + Θ r Y r σ s ( Y s ) Y s + Y r Y r σ s ( Y s ) + − Y s − Y s Y r Y r Y s )+ δ ir Θ r Θ s Y r + δ ir δ ir δ ir σ s (Θ s ) Y r Y r ]Θ s + ( δ ir δ is Θ s Y r Y s + δ ir δ is Θ s Y r Y s )Θ r + δ ir δ is Θ r Θ s Y r Y s + δ i r δ i s ( (3 − Y s − Y s )Θ s Y r Y r + Θ s Y r Y r Y s + ( Y r + Y s )Θ r Y r Y s Y s + Y r Y s ) } ǫ ( i )Similarly, we can show that(Ψ s Ψ r ) ǫ ( i )= { (Θ s Θ r ) − δ is Y s (Θ r Θ s ) Θ r − δ ir Y r (Θ s Θ r ) Θ s + δ ir δ is Y r Y s (Θ r Θ s ) + δ ir δ is Y r Y s (Θ s Θ r ) − ( δ ir δ is Y s Y s + δ ir Θ s Y r )Θ r Θ s Θ r − ( δ ir δ is (3 − Y s + Y s ) Y r Y r + δ is Θ r Y s )Θ s Θ r Θ s + δ i r [ δ i s ( − Y s Y r Y s + Θ r Y s Y s ) + δ ir Θ s Y r Y r ]Θ r Θ s + δ i r [ δ i r ( Θ s σ s σ r ( Y r ) Y r + σ s (Θ s ) Y r Y r ) + δ i s ( σ s σ r (Θ r ) σ s ( Y s ) Y s + σ s σ r ( Y r ) Y s + Y r σ s ( Y s ) Y s + − Y s Y r Y s Y s + Θ r Y s Y s )]Θ s Θ r − [ δ i r δ i s ( (2 − Y s )Θ s Y r Y s + Θ s σ r ( Y r ) Y r Y s + Θ s Y r Y r Y s + σ r (Θ r ) Y s Y s + σ r ( Y r ) Y s Y s + Y r Y s Y s ) + δ is Θ r Θ s Y s ]Θ r − [ δ i r δ i s ( ( Y r + Y s )Θ r Y r σ s ( Y s ) Y s + Y r σ s ( Y s ) + Θ r Y r Y s Y s + − Y s − Y s Y r Y r Y s + Θ r Y r Y s Y s ) + δ ir Θ s Θ r Y r + δ ir δ ir δ ir σ s (Θ s ) T r Y r ]Θ s + [ δ ir δ is Θ r Y r Y s + δ ir δ is Θ r Y r Y s ]Θ s + δ ir δ is Θ s Θ r Y r Y s + δ i r δ i s [ ( Y r + Y s )Θ r Y r Y s Y s + (3 − Y s − Y s )Θ s Y r Y r + Y r Y s + Θ s Y r Y r Y s ] } ǫ ( i )Using relations (4.34), (4.35) and (4.37), we obtain that[(Ψ r Ψ s ) − (Ψ s Ψ r ) ] ǫ ( i )= { ( δ ir δ is (1 − Y s ) Y s Y s Y s + δ ir Θ s − Θ s Y r )Θ r Θ s Θ r + ( δ ir δ is (3 − Y s + Y s )( Y r − Y r Y r Y r + δ is (Θ r − Θ r ) Y s )Θ s Θ r Θ s + δ i r [ δ i s ( − Y s − Y s σ r ( Y r ) Y r Y s + (1 − Y s ) Y r Y s − (2 − Y s ) Y r σ r ( Y r ) Y r Y s ) + δ is σ r (Θ r ) − Θ r Y s Y s + δ ir ( σ r σ s (Θ s ) − Θ s ) σ r ( Y r ) Y r ]Θ r Θ s + δ i r [ δ i s ( ( Y s + Y s − Y s Y r Y s Y s − (1 − Y r ) Y r Y s − (1 − Y s ) Y r Y s Y s ) + δ ir (Θ s − σ s (Θ s )) Y r Y r + δ is (1 − Y s )( σ s σ r (Θ r ) − Θ r ) Y s Y s ]Θ s Θ r + [ δ i r δ i s ( (1 − Y s )Θ s Y r Y s Y s + (2 − Y s )Θ s Y r Y s + ( Y r + Y s − s Y r Y s − Θ s Y r Y s Y s + (1 − Y s ) Y r σ r (Θ r ) Y s Y s + − Y s Y s Y s + (1 − Y r )(1 − Y s ) Y r Y s Y s ) + δ is (Θ r Θ s − Θ r Θ s ) Y s ]Θ r + [ δ i r δ i s ( ( Y r + Y s )Θ r Y r σ s ( Y s ) Y s + (2 − Y s )Θ r Y r Y s Y s − (2 − Y s )Θ r Y r Y s + ( Y r − Y r )Θ r Y r Y r Y s Y s + Y r − Y r Y r Y r σ s ( Y s ) + ( Y r − Y r )(3 − Y s − Y s ) Y r Y r Y s ) + δ ir δ ir ( Y r − Y r ) σ s (Θ s ) Y r Y r + δ ir (Θ r Θ s − Θ r Θ s ) Y r ]Θ s } ǫ ( i ) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 29 = δ i r Θ s − Θ s Y r Θ r Θ s Θ r ǫ ( i ) if i r = 0 , i r = 0 , i s = 0 ,δ i s Θ r − Θ r Y s Θ s Θ r Θ s ǫ ( i ) if i s = 0 , i r = 0 , i r = 0 ,δ i s Θ r Θ s − Θ r Θ s Y s Θ r ǫ ( i ) if i s = 0 , i r = 0 ,δ i r Θ r Θ s − Θ r Θ s Y r Θ s ǫ ( i ) if i r = 0 , i r = 0 ,δ i s Θ r Θ s − Θ r Θ s Y s Θ r ǫ ( i ) if i s = 0 , i r = 0 , i s = 0 ,δ i r Θ r Θ s − Θ r Θ s Y r Θ s ǫ ( i ) if i r = 0 , i s = 0 , i r = 0 , ( δ i r Θ s − Θ s Y r Θ r Θ s Θ r + δ i s Θ r − Θ r Y s Θ s Θ r Θ s ) ǫ ( i ) if i r = 0 , i s = 0 , i r = 0 , ( δ i r θ s − θ s y r θ r θ s θ r + δ i r δ i r σ r σ s ( θ s ) − θ s σ r ( Y r ) Y r Θ r Θ s + δ i r δ i r Θ s − σ s (Θ s ) Y r Y r Θ s Θ r + δ i r δ i r ( Y r − Y r ) σ s (Θ s ) Y r Y r Θ s + δ i r Θ r Θ s − Θ r Θ s Y r Θ s ) ǫ ( i ) if i r = 0 , i r = 0 , i r = 0 , i s = 0 , FAN KONG AND ZHI-WEI LI = Y s − Ψ r Ψ s Ψ r ǫ ( i ) if i r = − , i s = 1 , e = 2 , , − Y s Ψ r Ψ s Ψ r ǫ ( i ) if i r = 3 , i s = − , e = 2 , , − Y s + Y s − Y r Ψ s Ψ r Ψ s ǫ ( i ) if i r = − i s = 1 , e = 2 , − Y s + Y s Y r − Ψ s Ψ r Ψ s e ( i ) ǫ ( i ) if i r = − i s = − , e = 2 , (2 − Y s )( Y s − Y s + Y r Y s )(1 − Y r )(1 − Y s ) Ψ r ǫ ( i ) if i s = 1 , i r = − , e = 2 , ( Y s − Y s − Y s + Y r Y s )(1 − Y r )(1 − Y s ) Ψ r ǫ ( i ) if i s = − , i r = 2 , e = 2 , Y s ( Y s − − Y s Ψ s ǫ ( i ) if i r = 1 , i s = 2 , e = 4 , Y s ( Y s − − Y r Ψ s ǫ ( i ) if i r = − , i s = 2 , e = 4 , − Y s Ψ s ǫ ( i ) if i r = 3 , i s = − , e = 2 , , Y s − Ψ s ǫ ( i ) if i r = − , i s = 2 , e = 2 , , − Y r Ψ s ǫ ( i ) if i r = 1 , i s = − , e = 2 , , Y r − Ψ s ǫ ( i ) if i r = − , i s = 2 , e = 2 , , ( Y s − Y r Y s + Y r Y s (1 − Y r )](1 − Y s )(1 − Y r ) ( Y r Ψ r + 1) ǫ ( i ) if i r = 0 , i s = 1 , e = 2 , (2 − Y s )[ Y r Y s + Y r Y s (1 − Y s ) ](1 − Y r )(1 − Y s ) ( Y s ψ s + 1) ǫ ( i ) if i r = 1 , i s = 0 , e = 2 , [ Y s − Y s Ψ r Ψ s Ψ r − Y r (3 − Y s + Y s )1 − Y r Ψ s Ψ r Ψ s + Y r − Y r + Y s − Y s + Y s Y r (1 − Y s )(1 − Y r ) ] ǫ ( i ) if i r = i s = 1 , e = 2 , Y s − Ψ r Ψ s Ψ r ǫ ( i ) if i r = 0 , i s = 1 , e = 3 , − Y s (Ψ r Ψ s Ψ r + Ψ r Ψ s ) ǫ ( i ) if i r = 0 , i s = − , e = 3 , . For each w ∈ W , we fix a reduced decomposition w = σ r σ r · · · σ r m and definethe element Ψ w := Ψ r Ψ r · · · Ψ r m ∈ L q . Similar to the degenerate case, we can show that { Ψ w | w ∈ W} is a basis of L q as k ( Y ) ⊗ k E -module. Thus the relations (1)-(9) is complete since by them everyelement in L q can be written as P w ∈W , i ∈C Ψ w f w, i ( Y ) ǫ ( i ) with f w, i ( Y ) ∈ k ( Y ). (cid:3) The BK-subalgebras. We define the BK-subalgebra as the k -algebra ˜ L q gen-erated by { Y , · · · , Y n , Ψ , · · · , Ψ n , f − ( Y ) , ǫ ( i ) | i ∈ C , f ( Y ) ∈ k [ Y ] with f (0) = 0 } subject to the relations (1)-(10) of Theorem 4.38.Similar to the proof of Corollary 3.42, we have the following result. HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 31 Corollary 4.39. Denote by ǫ ( i ) = ǫ ( i ) . Then the algebra ˜ L q is generated by { X , · · · , X n , T , · · · , T n , f − ( X ) ǫ ( i ) | i ∈ C , f ( X ) ∈ k [ X ] with f ( q i ) = 0 } subject to relations (4.1)-(4.7), (3.20), (4.20), (4.22) and ǫ ( j ) · f − ǫ ( i ) = δ ji f − ǫ ( i ) = f − ǫ ( i ) · ǫ ( j ) , f · f − ǫ ( i ) = ǫ ( i ) = f − ǫ ( i ) · f (4.40) for f ∈ k [ X ] with f ( q i ) = 0 . Denote by ˜ L q (Λ) := ˜ L q / h Y Λ i ǫ ( i ) | i ∈ Ci . We use the same letters Ψ , · · · , Ψ n and Y , · · · , Y n to denote the images of thegenerators in ˜ L q (Λ). Imitate the proof of Lemma 3.44, we can deduce the followingresult. Lemma 4.41. ˜ L q (Λ) = ˜ L q / h Q i ∈ I ( X − q i ) Λ i i . Remark . Similar to the degenerate case, the elements Y r , Q i ∈ I ( X r − q i ) are allnilpotent in ˜ L q (Λ).4.8. The cyclotomic non-degenerate affine Hecke algebras. We define the non-degenerate cyclotomic affine Hecke algebra H q (Λ) as H q (Λ) := H q / h Y i ∈ I ( X − q i ) Λ i i . Similar to [3, Subsection 4.1], there is a system { e ( i ) | i ∈ C} of mutually orthogonalidempotents in H q (Λ) such that 1 = P i ∈ I n e ( i ) and e ( i ) H q (Λ) = { h ∈ H q (Λ) | ( X r − q i r ) m h = 0 for all r ∈ [ n ] and m ≫ } . It is easy to see that X r e ( i ) = e ( i ) X r for all r ∈ [ n ] and i ∈ C , and for a polynomial f ( X ) ∈ k [ X ], f ( X ) e ( i ) is a unit if and only if f ( q i ) = 0. In particular, the element(1 − X r ) e ( i ) is a unit in e ( i ) H q (Λ) if and only if i r = 0. In this case, we write(1 − X r ) − e ( i ) for the inverse. Lemma 4.43. For r ∈ [ n ] and i ∈ I n , there holds that T r e ( i ) = e ( i ) T r if i r = 0 ,e ( σ r ( i )) T r + − q − X r e ( σ r ( i )) − − q − X r e ( i ) if i r = 0 . (4.44) FAN KONG AND ZHI-WEI LI Proof. For any s ∈ [ n ], the element( σ r ( X s ) − q σ ( i ) s ) e ( i ) = [ X − a sr r ( X s − q i s ) + ( X r − q i r ) f a sr ( X ± r )] e ( i )for some polynomial f a sr ( X ± r ). Thus it is nilpotent by the nilpotency of ( X s − q i s ) e ( i ) and ( X r − q i r ) e ( i ). Similarly, we can show that D r (( X s − q i s ) m ) e ( i ) isnilpotent an integer m ≫ i r = 0. Therefore, if i r = 0, by (4.8), thereholds that ( X s − q i s ) m T r e ( i )= T r ( σ r ( X s ) − q i s ) m e ( i ) + (1 − q )D r (( X s − q i s ) m ) e ( i )= 0when m ≫ 0. Therefore T r e ( i ) ∈ e ( i ) H q (Λ) and then T r e ( i ) = e ( i ) T r e ( i ) = e ( i ) T r . If i r = 0, by (4.8), we get( X s − q σ r ( i ) s ) m [ T r (1 − X r ) + 1 − q ] e ( i )= [ T r (1 − X r ) + 1 − q ]( σ r ( X s ) − q σ r ( i ) s ) m e ( i )= 0when m ≫ 0. Therefore T r (1 − X r ) e ( i ) + (1 − q ) e ( i ) = e ( σ r ( i ))[ T r (1 − X r ) e ( i ) + (1 − q ) e ( i )]= e ( σ ( i )) T r (1 − X r ) e ( i ) . Then right-multiplying by (1 − X r ) − e ( i ), we have T r e ( i ) = e ( σ r ( i )) T r e ( i ) − (1 − q )(1 − X r ) − e ( i ) . Similarly, we can deduce that e ( σ r ( i )) T r = e ( σ r ( i )) T r e ( i ) − (1 − q )(1 − X r ) − e ( σ r ( i )) . Therefore T r e ( i ) = e ( σ r ( i )) T r + − q − X r e ( σ r ( i )) − − q − X r e ( i ) . (cid:3) Let e ( C ) := X i ∈C e ( i ) ∈ H q (Λ) . Then e ( C ) is a central idempotent in H q (Λ) by (4.44). Furthermore, by Lemma 4.41,(4.44) and Corollary 4.39, there is a homomorphism ρ q : ˜ L q (Λ) → H q (Λ) e ( C ) HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 33 sending the generators X r , T r to the same named elements, and f ( X ) − ǫ ( i ) with f ( q i ) = 0 to f ( X ) − e ( i ). Similar to the proof of Theorem 3.48, we arrive at oursecond main result in this paper for non-degenerate affine Hecke algebras. Theorem 4.45. There is an algebra isomorphism ˜ L q (Λ) ∼ = H q (Λ) e ( C ) . Generalization In this section, we give the general and unified definition for the KLR type algebrasin the previous Sections.5.1. The uniform quivers. Let I be an abelian group. We consider the loop-freequiver Γ I with vertex set I . Denote by d ij the number of arrows i → j . Then Γ I issaid to be a uniform quiver if d ij = d i ′ j ′ whenever i − j = i ′ − j ′ . For a fixed I , auniform quiver Γ I corresponds to a map from I \ { } to N .For the cyclic group I = Z /e Z = { , · · · , e − } , where e = 0 or 2 ≤ e ∈ Z , thequivers of type A ∞ if e = 0 or A (1) e − if e ≥ A ∞ : · · · − − · · · A (1) e − : 0 1 2 0 1 30 12 34 0 12 · · · and so are the followings: 413 02 50 1 Z ⊕ Z : (0 , 0) (1 , , 1) (1 , , 0) (1 , , 1) (1 , , 0) (1 , , 1) (1 , . The root system and the group map. Denote by [ n ] = { , , · · · , n } . LetΦ be a root system with a simple root system ∆ = { α r | r ∈ [ n ] } . Assume that W is the Weyl group of Φ. Assume W is generated by { σ r | r ∈ [ n ] } where σ r is thecorresponding simple reflection σ r of α r . Let R be a commutative associative unital FAN KONG AND ZHI-WEI LI k -algebra such that W has an action on it. We fix a W -map y : Φ → R, α y α that is, it is a map satisfies y ( w ( α )) = w ( y α ) for each w ∈ W . We assume y α is nota zero-divisor of R for each α ∈ Φ. Therefore R is a subalgebra of R Im y which is thelocalization of R with respect to Im y . We define the divided difference operators ∂ r on R to be ∂ r ( a ) = σ r ( a ) − ay α r . (5.1)The W -action on R extends to an action of W on R Im y via ring homomorphism. Thismeans that the action of the divided difference operators also extends to operatorson R Im y .5.3. The index set. Let S be a nonempty set. Denote by S ∗∗ = { f : S → I | f is a map } . Suppose there is an action of W on S ∗ . Then the W -actionon S ∗ extends to actions on S ∗∗ = { h : S ∗ → I | h is a map } by defining w ( h )( f ) = h ( w − ( f ))for w ∈ W , h ∈ S ∗∗ and all f ∈ S ∗ .Form now on, we fix a W -orbit C of S ∗ and an element η r in S ∗∗ for each r ∈ [ n ]. For simplicity, we denote by η r ( i ) := i r , η r ( i ) = σ s ( η r )( i ) := i r , η r ( i ) = σ r σ s ( η r )( i ) := i r for each i ∈ C .5.4. Coproducts of associated algebras. Let k be a commutative ring. Recallin [2, Subsection 1.4], given two k -algebras A and A with 1, the coproduct A ⊔ k A of A and A is defined to be the quotient of the tensor algebra T ( A ⊕ A ) = A ⊕ A ⊕ A ⊗ A ⊕ A ⊗ A ⊕ A ⊗ A ⊕ · · · modulo the ideal generated by all elements of the form a ⊗ b − a b , a ⊗ b − a b , A − A where a , b ∈ A , a , b ∈ A . HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 35 A KLR-type algebra. Let Γ I be an uniform quiver with vertex set I . Foreach m ∈ I , we fix a W -map L m : Im y → R Im y which is decided by L m ( y α ) = y − α + σ r ( y α ) − if m = 0; y d ij α σ r ( y α ) d ji if ∃ i = j ∈ I such that m = i − j (5.2)for α ∈ Φ. Denote by L = ( L m ) m ∈ I . For simplicity, we denote by y r = y α r , y r = y σ αs ( α r ) , y r = y σ αr σ αs ( α r ) for r ∈ [ n ]. Definition 5.3. Let A be a unital k -algebra generated by { ǫ ( i ) , ψ r | i ∈ C , r ∈ [ n ] } .The algebra R (Γ I , C , R, y ) is defined to be the coproduct A ⊔ k R together with thefollowing relations for all admissible indices:(1) P i ∈C ǫ ( i ) = 1 , ǫ ( i ) ǫ ( j ) = δ ij ǫ ( i );(2) aǫ ( i ) = ǫ ( i ) a, ∀ a ∈ R ;(3) ψ r ǫ ( i ) = ǫ ( σ r ( i )) ψ r ;(4) ψ r y s ǫ ( i ) = σ r ( y s ) ψ r ǫ ( i ) + δ i r ∂ r ( y s ) ǫ ( i );(5) L ( y r ) ∈ R and ψ r ǫ ( i ) = − ψ r L ( y r ) ǫ ( i ) if i r = 0 ,L i r ( y r ) ǫ ( i ) if i r = 0;(6) if r / − s , then ψ r ψ s = ψ s ψ r ;(7) if r − s , then( ψ r ψ s ψ r − ψ s ψ r ψ s ) ǫ ( i ) = δ i s L i s ( y s ) − L i r ( y r ) y s ǫ ( i );(8) if r i s , then[( ψ r ψ s ) − ( ψ s ψ r ) ] ǫ ( i )= [ δ i r L i s ( y s ) − L i s ( y s ) y r ψ r − δ i s L i r ( y r ) − L i r ( y r ) y s ψ s + δ i r δ i r L i s ( y s ) − L i s ( y s ) y r y r ] ǫ ( i );(9) if r i s , then[( ψ r ψ s ) − ( ψ s ψ r ) ] ǫ ( i )= [ δ i r L i s ( y s ) − L i s ( y s ) y r ψ r ψ s ψ r + δ i s L i r ( y r ) − L i r ( y r ) y s ψ s ψ r ψ s FAN KONG AND ZHI-WEI LI + δ i r δ i r y r L i s ( y s ) + ( y r − y r ) L i s ( y s ) − y r L i s ( y s ) y r y r y r ψ r ψ s + δ i r L i r ( y r ) L i s ( y s ) − L i r ( y r ) L i s ( y s ) y r ψ s + δ i s L i r ( y r ) L i s ( y s ) − L i r ( y r ) L i s ( y s ) y s ψ r + δ i s δ i r L i r ( y r ) L i s ( y s ) − L i r ( y r ) L i s ( y s ) y s y r + δ i r δ i s L i s ( y s ) L i r ( y r ) − L i s ( y s ) L i r ( y r ) y r y s + δ i r δ i s L i s ( y s ) L i r ( y r ) − L i r ( y r ) L i s ( y s ) y r y s ] ǫ ( i ) . (10) all the coefficients in (7)-(9) are in R . Remark . (a) If R = R Im y , then the condition (10) holds automatically. Forexample, the Lusztig extension in KLR form.In the sequel, we take I = Z /e Z and Γ I = A ∞ or A (1) e − .(b) Take R = k [ y , · · · , y n ] to be the polynomial ring with indeterminates { y r | r ∈ [ n ] } and k ( y , · · · , y n ) the corresponding rational functions field. Then there is anaction of W on R (by ring automorphism) such that for every r ∈ [ n ] σ r ( y s ) = y s − a sr y r . Let y : Φ → R be the map by sending P r ∈ [ n ] a r α r to P r ∈ [ n ] a r y r . Then y is a W -mapand the map L m : Im y → k ( y , · · · , y n ) in (5.2) is given by L m ( y r ) = m = 0; − y r if m = 1 , e = 2; y r if m = − , e = 2; − y r if m = 1 , e = 2;1 if m = 0 , ± . Take S = { , , · · · , n } , then S ∗ = I n and there is a W -action on I n as defined by(2.6). Let η r be the map η r ( i ) = i r for all i ∈ I n and r ∈ [ n ]. Then the algebra R (Γ I , C , R, y ) is isomorphic to the algebra R as defined in Subsection 3.8.(c) Let R = { fg | f, g ∈ k [ y , · · · , y n ] with g (0) = 0 } ⊂ k ( y , · · · , y n ). Then thereis an action of W on R (via the ring automorphism) induced by σ r ( y s ) = 1 − (1 − y s )(1 − y r ) − a sr . HE BRUNDAN-KLESHCHEV SUBALGEBRA AND BK-TYPE ISOMORPHISM 37 Let y : Φ → R be the map decided by sending P r ∈ [ n ] a r α r to Q r ∈ [ n ] (1 − y r ) a r . then y is a W -map and the map L m : Im y → K ( y , · · · , y n ) in (5.2) is given by L m ( y r ) = m = 0; y r y r − if m = 1 , e = 2; y r if m = − , e = 2; y r y r − if m = 1 , e = 2;1 if m = 0 , ± . Take S = { , , · · · , n } , then S ∗ = I n and there is a W -action on I n as defined by(2.6). Let η r be the map η r ( i ) = i r for all i ∈ I n and r ∈ [ n ]. In this case, the algebra R (Γ I , C , R, y ) is isomorphic to the BK subalgebra ˜ L q as defined in Subsection 4.7.(d) Take W = S n to be the symmetric group. Let R = k [ y , · · · , y n +1 ] be thepolynomial ring, then there is an action of S n on R by permuting variables. Let y : Φ → R be the map by sending P r ∈ [ n ] a r α r to P r ∈ [ n ] a r ( y r − y r +1 ). Then y is a W -map and the map L m : Im y → k ( y , · · · , y n +1 ) in (5.2) is given by L m ( y r ) = m = 0; y r +1 − y r if m = 1 , e = 2; y r − y r +1 if m = − , e = 2;( y r +1 − y r )( y r − y r +1 ) if m = 1 , e = 2;1 if m = 0 , ± . Take S = { , , · · · , n + 1 } , then S ∗ = I n +1 and there is an S n -action on S ∗ bythe place permutation : w ( i ) s = i w − ( s ) . Let η r be the map η r ( i ) = i r − i r +1 for all i ∈ I n +1 and r ∈ [ n ]. Then the algebra R (Γ I , C , R, y ) is isomorphic to KLR algebraof type A as defined in [6, Subsection 4.4]. References [1] A. Bj¨orner and F. Brenti, Combinatorics of Coxeter groups, Springer GTM231, 2005.[2] K. I. Beida, W. S. Matindale III and A. V. Mikhalev, Rings with generalized identities,Monographs and textbooks in pure and applied mathematics , 1995.[3] J. Brundan and A. Kleshchev, Blocks of cyclotomic Hecke algebras and Khovanov-Laudaalgebra. Invent. math. (3) (2009), 451-484.[4] J. Brundan, A. Kleshchev and Weiqiang Wang, The graded Specht modules. J. reine angew.Math. (2011), 61-87. FAN KONG AND ZHI-WEI LI [5] V. Drinfeld, Degenerate affine Hecke algebras and Yangians, Func. Anal. Appl. (1986),56-58.[6] Fan Kong and Zhi-Wei Li, The Brundan-Kleshchev isomorphism revisited, arXiv:2102.06473.[7] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. (1989),599-635. Fan Kong, School of Mathematics and Statistics, Southwest University, Chongqing400715, PR China. Email address : [email protected] Zhi-Wei Li, School of Mathematics and Statistics, Jiangsu Normal University,Xuzhou 221116 Jiangsu, PR China. Email address ::