For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, ...In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.展开更多
Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig poly...Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.展开更多
Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra ac...Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.展开更多
基金supported by National Key R&D Program of China (Grant No. 2020YFA0712600)National Natural Science Foundation of China (Grant No. 11688101)the AMSS for hospitality and for financial supports。
文摘For type B_(3), we show that Lusztig's conjecture on the structure of the based ring of the two-sided cell corresponding to the unipotent class in Sp_(6)(C) with three equal Jordan blocks needs modification.
文摘In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.
基金Supported in part by the Natural Science Foundation of China(Grant no.11101233)Beijing Youth Top-notch Talent Support Program(Grant no.21351918007).
文摘Let(W,S)be the affine Weyl group of type B_(2),on which we consider the length function e from W to N and the Bruhat order≤.For y<w in W,letμ(y,w)be the coefficient of q^(1/2(e(w)-e(y)-1)) in Kazhdan-Lusztig polynomial P_(y,w)∈Z[q].We determine someμ(y,w)for y∈c_(0) and w∈c_(2),where c0 is the lowest two-sided cell of B_(2) and c_(2) is the higher one.Furthermore,we get some consequences using left or right strings and some properties of leading coefficients.
基金supported by National Natural Science Foundation of China(Grant No.11326059)
文摘Given a suitable ordering of the positive root system associated with a semisimple Lie algebra,there exists a natural correspondence between Verma modules and related polynomial algebras. With this, the Lie algebra action on a Verma module can be interpreted as a differential operator action on polynomials, and thus on the corresponding truncated formal power series. We prove that the space of truncated formal power series gives a differential-operator representation of the Weyl group W. We also introduce a system of partial differential equations to investigate singular vectors in the Verma module. It is shown that the solution space of the system in the space of truncated formal power series is the span of {w(1) | w ∈ W }. Those w(1) that are polynomials correspond to singular vectors in the Verma module. This elementary approach by partial differential equations also gives a new proof of the well-known BGG-Verma theorem.
