BGG duality and Macdonald Polynomials Vyjayanthi Chari University of California, Riverside Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 1 / 10
I am interested in the representation theory of affine and quantum affine algebras. In particular, I am trying to understand the homological properties of the (non–semisimple) category of finite–dimensional representations of these algebras. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 2 / 10
I am interested in the representation theory of affine and quantum affine algebras. In particular, I am trying to understand the homological properties of the (non–semisimple) category of finite–dimensional representations of these algebras. It turns out that some of these problems are closely related to the analogous problems for a particular maximal parabolic subalgebra of the affine Lie algebra: namely the one obtained by dropping the zero–th node of the affine Dynkin diagram. In the case of the untwisted affine Lie algebra, this parabolic subalgebra is called the current algebra and is easily described as follows. Let g be the underlying simple finite–dimensional Lie algebra. The current algebra is just g [ t ] = g ⊗ C [ t ] with the obvious commutator, [ x ⊗ t r , y ⊗ t s ] = [ x , y ] ⊗ t r + s . Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 2 / 10
Today, I want to explain the connection between the representation theory of the current algebra and the following combinatorial identity. This is based on joint work with my former students Matthew Bennett and Nathan Manning, [BCM] and also on joint work with Bennett, Berenstein, Khoroskhin and Loktev [BBCKL]. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 3 / 10
Today, I want to explain the connection between the representation theory of the current algebra and the following combinatorial identity. This is based on joint work with my former students Matthew Bennett and Nathan Manning, [BCM] and also on joint work with Bennett, Berenstein, Khoroskhin and Loktev [BBCKL]. Given a partition µ = ( µ 1 ≥ · · · ≥ µ r − 1 ≥ 0), let P µ ( x 1 , · · · , x r ; q , 0) be the specialized Macdonald polynomial and write it as, η µ � � P µ ( x 1 , . . . , x r ; q , 0) = λ,ℓ ( q ) s λ,ℓ ( x 1 , . . . , x r ) , ℓ ≥ 0 λ ∈ Par ( r − 1) where s λ,ℓ = | x 1 · · · x r | ℓ s λ ( x 1 , · · · , x r ) and s λ is the Schur function associated to λ . Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 3 / 10
Today, I want to explain the connection between the representation theory of the current algebra and the following combinatorial identity. This is based on joint work with my former students Matthew Bennett and Nathan Manning, [BCM] and also on joint work with Bennett, Berenstein, Khoroskhin and Loktev [BBCKL]. Given a partition µ = ( µ 1 ≥ · · · ≥ µ r − 1 ≥ 0), let P µ ( x 1 , · · · , x r ; q , 0) be the specialized Macdonald polynomial and write it as, η µ � � P µ ( x 1 , . . . , x r ; q , 0) = λ,ℓ ( q ) s λ,ℓ ( x 1 , . . . , x r ) , ℓ ≥ 0 λ ∈ Par ( r − 1) where s λ,ℓ = | x 1 · · · x r | ℓ s λ ( x 1 , · · · , x r ) and s λ is the Schur function associated to λ . Then, one can prove [BBCKL] that: η µ λ,ℓ ( q ) P µ (1 , · · · , 1; q , 0) s λ (1 , . . . , 1) � � = . ( q ; q ) r 2 − 1 ( q ; q ) µ 1 − µ 2 . . . ( q ; q ) µ r − 2 − µ r − 1 ( q ; q ) µ r − 1 ∞ ℓ ≥ 0 µ ∈ Par ( r − 1) Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 3 / 10
Graded Representations The current algebra has an obvious grading by the non–negative integers, we just declare an element x ⊗ t r to have grade r . Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 4 / 10
Graded Representations The current algebra has an obvious grading by the non–negative integers, we just declare an element x ⊗ t r to have grade r . So, we work in the category G of Z –graded representations of g [ t ] with morphisms being degree zero maps. We also require that all graded components be finite–dimensional. In particular, objects of G regarded as g –modules are completely reducible. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 4 / 10
Graded Representations The current algebra has an obvious grading by the non–negative integers, we just declare an element x ⊗ t r to have grade r . So, we work in the category G of Z –graded representations of g [ t ] with morphisms being degree zero maps. We also require that all graded components be finite–dimensional. In particular, objects of G regarded as g –modules are completely reducible. This makes it easy to describe the irreducible objects in the category: just put an irreducible finite–dimensional g –module in a particular grade and make g ⊗ t C [ t ] act trivially on it. Lemma The isomorphism classes irreducible objects of G are indexed by pairs ( λ, r ) where λ is a dominant integral weight of g and r ∈ Z . Denote by V ( λ, r ) a representation in the corresponding isomorphism class. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 4 / 10
Projective modules Set P ( λ, r ) = U ( g [ t ]) ⊗ U ( g ) V ( λ, r ) . A standard proof shows that it is a projective infinite–dimensional object of G . A simple application of the PBW theorem shows that the Hilbert series of P ( λ, r ) is H ( P ( λ, r )) = q r dim V ( λ ) H ( S ( g [ t ] + )) = dim V ( λ ) . ( q ; q ) dim g ∞ In the case when g is of type sl r , the dimension of V ( λ ) is exactly s λ (1 , · · · , 1) and so we see the left hand side of the combinatorial identity. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 5 / 10
Local and Global Weyl modules and Demazure modules The global Weyl module W ( λ, r ) is the maximal quotient of P ( λ, r ) whose weights lie below λ , i.e., are in λ − Q + . It is infinite–dimensional. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 6 / 10
Local and Global Weyl modules and Demazure modules The global Weyl module W ( λ, r ) is the maximal quotient of P ( λ, r ) whose weights lie below λ , i.e., are in λ − Q + . It is infinite–dimensional. The local Weyl module W loc ( λ, r ) is the maximal quotient of W ( λ, r ) such that the λ weight space is one–dimensional. These are finite–dimensional, indecomposable and usually reducible modules. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 6 / 10
Local and Global Weyl modules and Demazure modules The global Weyl module W ( λ, r ) is the maximal quotient of P ( λ, r ) whose weights lie below λ , i.e., are in λ − Q + . It is infinite–dimensional. The local Weyl module W loc ( λ, r ) is the maximal quotient of W ( λ, r ) such that the λ weight space is one–dimensional. These are finite–dimensional, indecomposable and usually reducible modules. These modules were originally defined in 2000, in joint work with Pressley.The local Weyl modules are actually q = 1 limits of irreducible representations of the quantum affine algebra. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 6 / 10
In the simply laced case, the local Weyl modules are actually a special family of Demazure modules. Namely, those which occur in level one representations of the affine Lie algebra and which admit a g –module structure. This was proved for sl n +1 in joint work with Loktev in 2006 and in 2007 by Fourier and Littelmann for A , D , E . Both these papers used in an important way, the results established for sl 2 in the work with Pressley. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 7 / 10
In the simply laced case, the local Weyl modules are actually a special family of Demazure modules. Namely, those which occur in level one representations of the affine Lie algebra and which admit a g –module structure. This was proved for sl n +1 in joint work with Loktev in 2006 and in 2007 by Fourier and Littelmann for A , D , E . Both these papers used in an important way, the results established for sl 2 in the work with Pressley. Hence, for sl r , the graded character of the local Weyl module is given by P λ ( x ; q , 0). This is because Y.Sanderson had computed the graded character of such Demazure modules. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 7 / 10
The graded character of the global Weyl module W ( λ, r ) is, ch gr W ( λ, 0) = ch gr W loc ( λ, 0) H ( W ( λ, 0)) = H ( W loc ( λ, 0)) , , � n � n i =1 ( q ; q ) λ i i =1 ( q ; q ) λ i where λ = � n i =1 λ i ω i and ω i are the fundamental weights of g . This uses some deep results on global Weyl modules, for sl r it follows from [CP2000] and [CLoktev]. Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 8 / 10
The graded character of the global Weyl module W ( λ, r ) is, ch gr W ( λ, 0) = ch gr W loc ( λ, 0) H ( W ( λ, 0)) = H ( W loc ( λ, 0)) , , � n � n i =1 ( q ; q ) λ i i =1 ( q ; q ) λ i where λ = � n i =1 λ i ω i and ω i are the fundamental weights of g . This uses some deep results on global Weyl modules, for sl r it follows from [CP2000] and [CLoktev]. So for sl r , we have P λ ( x ; q , 0) ch gr W ( λ, 0) = . � n i =1 ( q ; q ) λ i So our original combinatorial identity now becomes an equality of Hilbert series, � � H ( W ( µ, 0))[ W loc ( µ, 0) : V ( λ, k )] q k . H ( P ( λ, 0)) = k ≥ 0 µ ∈ P + Vyjayanthi Chari (UCR) BGG duality and Macdonald Polynomials 8 / 10
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