Demazure module

In mathematics, a Demazure module, introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year1=1974a|year2=1974b}}, is a submodule of a finite-dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The Demazure character formula, introduced by {{harvs|txt|authorlink=Michel Demazure|last=Demazure|year=1974b|loc=theorem 2}}, gives the characters of Demazure modules, and is a generalization of the Weyl character formula.

The dimension of a Demazure module is a polynomial in the highest weight, called a Demazure polynomial.

Demazure modules

Suppose that g is a complex semisimple Lie algebra, with a Borel subalgebra b containing a Cartan subalgebra h. An irreducible finite-dimensional representation V of g splits as a sum of eigenspaces of h, and the highest weight space is 1-dimensional and is an eigenspace of b. The Weyl group W acts on the weights of V, and the conjugates wλ of the highest weight vector λ under this action are the extremal weights, whose weight spaces are all 1-dimensional.

A Demazure module is the b-submodule of V generated by the weight space of an extremal vector wλ, so the Demazure submodules of V are parametrized by the Weyl group W.

There are two extreme cases: if w is trivial the Demazure module is just 1-dimensional, and if w is the element of maximal length of W then the Demazure module is the whole of the irreducible representation V.

Demazure modules can be defined in a similar way for highest weight representations of Kac–Moody algebras, except that one now has 2 cases as one can consider the submodules generated by either the Borel subalgebra b or its opposite subalgebra. In the finite-dimensional these are exchanged by the longest element of the Weyl group, but this is no longer the case in infinite dimensions as there is no longest element.

Demazure character formula

=History=

The Demazure character formula was introduced by {{harv|Demazure|1974b|loc=theorem 2}}.

Victor Kac pointed out that Demazure's proof has a serious gap, as it depends on {{harv|Demazure|1974a|loc=Proposition 11, section 2}}, which is false; see {{harv|Joseph|1985|loc=section 4}} for Kac's counterexample. {{harvtxt|Andersen|1985}} gave a proof of Demazure's character formula using the work on the geometry of Schubert varieties by {{harvtxt|Ramanan|Ramanathan|1985}} and {{harvtxt|Mehta|Ramanathan|1985}}. {{harvtxt|Joseph|1985}} gave a proof for sufficiently large dominant highest weight modules using Lie algebra techniques. {{harvtxt|Kashiwara|1993}} proved a refined version of the Demazure character formula that {{harvtxt|Littelmann|1995}} conjectured (and proved in many cases).

=Statement=

The Demazure character formula is

: $\text{Ch}(F(w\lambda)) = \Delta_1\Delta_2\cdots\Delta_ne^\lambda$

Here:

w is an element of the Weyl group, with reduced decomposition w = s₁...s_n as a product of reflections of simple roots.
λ is a lowest weight, and e^λ the corresponding element of the group ring of the weight lattice.
Ch(F(wλ)) is the character of the Demazure module F(wλ).
P is the weight lattice, and Z[P] is its group ring.
$\rho$ is the sum of fundamental weights and the dot action is defined by $w\cdot u=w(u+\rho)-\rho$ .
Δ_α for α a root is the endomorphism of the Z-module Z[P] defined by

: $\Delta_\alpha(u) = \frac{u-s_\alpha \cdot u}{1-e^{-\alpha}}$

:and Δ_j is Δ_α for α the root of s_j

References

{{Citation | last1=Andersen | first1=H. H. | title=Schubert varieties and Demazure's character formula | doi=10.1007/BF01388527 | mr=782239 | year=1985 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=79 | issue=3 | pages=611–618| bibcode=1985InMat..79..611A | s2cid=121295084 }}
{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Désingularisation des variétés de Schubert généralisées | mr=0354697 | year=1974a | journal=Annales Scientifiques de l'École Normale Supérieure | issn=0012-9593 | volume=7 |series=Série 4 | pages=53–88| doi=10.24033/asens.1261 | doi-access=free }}
{{Citation | last1=Demazure | first1=Michel | author1-link=Michel Demazure | title=Une nouvelle formule des caractères | mr=0430001 | year=1974b| journal=Bulletin des Sciences Mathématiques |series=2e Série | issn=0007-4497 | volume=98 | issue=3 | pages=163–172}}
{{Citation | last1=Joseph | first1=Anthony | title=On the Demazure character formula | mr=826100 | year=1985 | journal=Annales Scientifiques de l'École Normale Supérieure |series=Série 4 | issn=0012-9593 | volume=18 | issue=3 | pages=389–419| doi=10.24033/asens.1493 | doi-access=free }}
{{Citation | last1=Kashiwara | first1=Masaki | author1-link=Masaki Kashiwara | title=The crystal base and Littelmann's refined Demazure character formula | doi=10.1215/S0012-7094-93-07131-1 | mr=1240605 | year=1993 | journal=Duke Mathematical Journal | issn=0012-7094 | volume=71 | issue=3 | pages=839–858}}
{{Citation | last1=Littelmann | first1=Peter | title=Crystal graphs and Young tableaux | doi=10.1006/jabr.1995.1175 | mr=1338967 | year=1995 | journal=Journal of Algebra | issn=0021-8693 | volume=175 | issue=1 | pages=65–87| doi-access=free }}
{{Citation | last1=Mehta | first1=V. B. | last2=Ramanathan | first2=A. | title=Frobenius splitting and cohomology vanishing for Schubert varieties | doi=10.2307/1971368 | mr=799251 | year=1985 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=122 | issue=1 | pages=27–40| jstor=1971368 }}
{{Citation | last1=Ramanan | first1=S. | last2=Ramanathan | first2=A. | title=Projective normality of flag varieties and Schubert varieties | doi=10.1007/BF01388970 | mr=778124 | year=1985 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=79 | issue=2 | pages=217–224| bibcode=1985InMat..79..217R | s2cid=123105737 }}

Category:Representation theory