### Research Results on Gelfand Kirillov Dimension of Hasish Chandra Module for Su(p,q) of Lie Groups by BIT

**Beijing Institute of Technology, Feb 5 ^{th}**, 2020: Recently, Xie Xun, associate researcher, School of Mathematics and Statistics, BIT, published a research paper entitled Gelfand–Kirillov Dimensions of Highest Weight Harish-Chandra Modules for SU(p,q) in International Mathematics Research Notices, a top international academic journal of mathematics. In this paper, an algorithm for the Gelfand-Kirillov dimension of the highest weighted single-module SU(p, q) is given, which can be used to study the Gelfand-Kirillov dimensions of these single-module and some properties of adjoint clusters.

Gelfand-Kirillov dimension is a mathematical concept proposed by Gelfand-Kirillov in 1966, which has become an important invariant for the representation theory of Lie Group Lie Algebra to measure the size of infinite dimensional modules. The research object is the Gelfand-Kirillov dimension of the highest weight Harish-Chandra module of the Hermitian type Lie Group SU(p,q), and the paper gives a concise algorithm for directly obtaining the Gelfand-Kirillov dimension from the highest weight. A basic tool applied is a formula briefly mentioned by Lusztig in his article in 1984, which gives a very beautiful formula of Gelfand Kirillov dimension of some highest weight modules and a-function of elements of Weyl group.

The Weyl group of type A algebra is a symmetric group, and its a-function can be given by the well-known RSK algorithm. The second idea of this paper is to get Yang graph directly by RSK algorithm, and then give a-function and Gelfand Kirillov dimension. In this way, the steps of finding the elements of the corresponding Weyl group by the highest weight can be omitted.

Since the highest weights of the Harish-Chandra modules with the highest weights of SU(p, q) are (p,q)-dominated, the paper focuses on the highest A-type (p,q)-dominated weight Gelfand-Kirillov dimension of the weights. The conclusion is that these weights correspond to a maximum of two columns of the Yang graph, so the Gelfand-Kirillov dimension of their corresponding highest weight module can be completely determined by the length of the second column corresponding to them, and a combinatorial model is given to explain how to read the length of the second column of the poplar directly from (p,q)-dominated highest weight, and then some properties of the Gelfand Kirillov dimension are obtained.

This research introduces a new method for the study of Gelfand Kirillov dimension and adjoint cluster, that is, the method of using a function of Hecke algebra. Through this method, some combined methods can be introduced to study the problems related to Gelfand Kirillov dimension and adjoint cluster. This provides an important reference value for the follow-up research work, which is a highlight of this article. The method of this paper can be extended to the study of other classical Lie Groups. There are many interesting problems worthy of further discussion. Especially on the research of BCD type a function combination algorithm, the related work is under study at present.

The research work which is funded by the Academic Initiation Program of BIT was completed by Xie Xun and Bai Zhanqiang, associate professor, Suzhou University, and Xie is the corresponding author.

**Paper link: **https://doi.org/10.1093/imrn/rnx247

**Research team and personal profile are attached as follows:**

The Algebra Team, School of Mathematics and Statistics, BIT, actively carries out international cooperative research and academic exchanges. Team leader Professor Hu Jun and his members Professor Wei Feng, Professor Wan Jinkui, Associate Professor Zhang Jie, Associate Professor Michael Ehrig, Dr. Xun Xun, etc. have cooperation with researchers of The University of Sydney, Universidade Federal do ABC, University of Virginia and University of Sherbrooke. Team members respectively carried out cross-researches of Representation Theory, Lie Theory, Algebraic Combination and Cluster Algebra, which shows a strong momentum of development.

Xie Xun, associate researcher, School of Mathematics and Statistics, BIT, is a member of Algebra Team. He graduated from the University of Chinese Academy of Sciences with a doctorate degree and did post-doctoral research at Peking University before visiting the University of Sydney for a year. Xie has long been engaged in the research of algebraic groups, quantum groups, and Hecke algebras. Currently, he has hosted the National Natural Science Foundation of China Youth Project and has published 4 papers in International Mathematics Research Notices, Journal of Algebra, Journal of Pure and Applied Algebra.

**Editor:** News Agency of BIT

**Translation: **News Agency of BIT