Weyl s complete reducibility theorem, levi and maltsev theorems. Example 352 the weyl denominator formula for the a. Cartan sub lie algebra, roots, weyl group, dynkin diagram. Through elementary combinatorics and noncanonical forms of wellknown lie algebras, we are able to. Introduction to avoid the encumbrance of technicalities, we present the results first for simple finitedimensional lie algebras over c. The author proceeds beyond the representation theory of compact lie groups which is the basis of many texts and provides a carefully chosen range of material to give the student the bigger picture.
The peter weyl theorem asserts that r occurs in l2uwithmultiplicity equaltothedegreeofr. The weyl character ring is the representation ring of a compact lie group. If v is a representation of a semisimple lie algebra l we have a homomorphism. Notes on weyl modules for semisimple algebraic groups october 24, 2014 over many decades of development, there has been some evolution in. Notes on the weyl character formula the aim of these notes is to give a selfcontained algebraic proof of the weyl character formula. The weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. The short third part will be devoted to the proof weyls character formula. Fundamental weights, permutation weights and weyl character. These can be derived from a recursive formula for characters valid for all simple complex kacmoody lie algebras for which the weyl kac character formula holds. Weyls character formula for representations of semisimple lie. Weylkac character formula for affine lie algebra in delignes category. A recursive formula for characters of simple lie algebras.
One of the most important new features of loop group representations is the modular invariance of the kac weyl character formula, which allows us to derive many interesting. To each finite dimensional irreducible representation v. The main novelty is the interpretation of the numerator in the weyl character formula in terms of intertwining operators. This is basically due to atiyah and bott in ab68, but we shall give our own presentation of their ideas in section 3. Proof of weyl character formula brandeis university. Using weyls formula one can conceivably compute the character of any semisimple lie group, at the expense of carrying out a complicated. The weyl algebras arise in a number of contexts, notably as a quotient of the universal enveloping algebra of certain. Classification of irreducible finitedimensional representations of semisimple lie algebras. This will lead into nitedimensional representations and the weyl character formula. There is a closely related formula for the character of an irreducible representation of a semisimple lie algebra. To each point of the ag variety bthere is associated a nilpotent subalgebra n of g, and takes a representation v to the alternating sum of the bundles on xwith lie algebra homology bers h n.
The weyl character formula, the halfspin representations. In 4, frenkel develops a generalization of kirillovs orbit. Euclidean space rn appear in lie theory as finite and affine weyl groups and play a central. If time permit we shall also see some connections with theta functions.
Let u be a compact connected semisimple lie group and t. Stanley 17 has observed that for finitedimensional simple lie algebras, this numerator formula, which was. Weyls character formula for nonconnected lie groups and. Pdf let b be a reductive lie subalgebra of a semisimple lie algebra f of the same rank both over the complex numbers. Let gbe a compact lie group with lie algebra g, tbe a maximal torus of gwith corresponding cartan subalgebra t. The condition that it is an eigendistribution can be relaxed slightly to the condition that its image under the center of the universal enveloping algebra is finitedimensional. The weyl character formula tells us how to go from the former to the latter. To get an action of the weyl group we used the fact that the. Some technical details are left to the exercises at the end. Loktev, on generalized kostka polynomials and the quantum verlinde rule, in. Weyl s character formula describes the character as the quotient of a sum over the weyl group and a product over positive roots. The weyl character formula calculates the restrictions to a maxima l.
Finally, the second edition adds a new section on the structure and representations of compact lie groups. The weyl character formula atiyahsinger index theorem abstract. The compleat weyl character formula boston college. Representations and invariants of the classical groups. I am not aware of a textbook where this is done, but i used humphreyss book 15 as reference for the lie algebra parts the general exposition and especially the proof of the weyl character formula, and the introduction. The character of the corresponding simple module ll is given by the weyl character formula 27,28,29. Finally we will apply this to the classi cation of compact lie groups. This proof is somewhat indirect and well describe later some other di erent proofs. The weyl denominator formula amounts to the character formula for the trivial 1dimensional module and is hard to make intuitive.
The weyl character formula math g4344, spring 2012 1 characters we have seen that irreducible representations of a compact lie group gcan be constructed starting from a highest weight space and applying negative roots. Differential topology, infinitedimensional lie algebras, and applications, in. Verma modules for infinite dimensional lie algebras are much the same as for finite dimensional ones. Weyl s character formula can be approached from many directions, so what is simple depends heavily on what you already know and what your further interests are. Lie groups and lie algebras for physicists harold steinacker lecture notes1, spring 2015 university of vienna fakult at fur physik universit at wien boltzmanngasse 5, a1090 wien, austria email. Lie groups, lie algebras, and their representations. Cartan decomposition on the lie algebra level 354 3. Let gbe a compact lie group and x a maximal torus t.
This is the first of two lectures on the representations of compact lie groups. Even more important than the character formula itself is a corollary, the weyl dimension formula, which gives a fairly simple expression for the total dimension of the representation in terms of the root system of the lie algebra. The second chapter of this paper covers some basic results on the weyl. Ive gone through a few low rank examples, and many of the coefficients are coming out to be zero. The weyl character formula, the halfspin representations, and equal. Weyl, demazure and fusion modules for the current algebra.
A representation of an associative algebra aalso called a left. For compact lie groups, the book covers the peter weyl theorem, lie algebra, conjugacy of maximal tori, the weyl group, roots and weights, weyl character formula. Although much of his working life was spent in zurich, switzerland and then princeton, new jersey, he is associated with the university of gottingen tradition of mathematics, represented by david hilbert and hermann minkowski. Let g c and t c be the complexi cations of g and t respectively. At the end of the 19th century killing and cartan classi ed the nite dimen. If time permits we may study lie algebras over the field of real numbers or look at jordan algebras. Characters, harmonic analysis, andan l2lefschetz formula jamesarthur. Weyls character formula is a useful tool in understanding the structure of irreducible representations of semisimple lie algebras. The simple objects are in bijection with weights in the fundamental alcove of level k. A lie algebra is a vector space with an antisymmetric product satisfying the jacobi identity. For any vector space v we have slv, the special linear lie algebra of v, consisting of the operators on v of trace 0.
Weyl s character formula is a useful tool in understanding the structure of irreducible representations of semisimple lie algebras. Weylkac character formula for affine lie algebra in deligne. For a semisimple lie algebra g, with triangular decomposition g n. Fundamental weights, permutation weights and weyl character formula h r karadayi and m gungormez department of physics, faculty of science, technical university of istanbul, 80626, maslak, istanbul, turkey received 24 june 1998 abstract.
It has a basis consisting of the irreducible representations of \g\, or equivalently, their characters. Universal enveloping algebra and the casimir operator. Intergrable representations of kacmoody lie algebras. Character formulas, dario beraldo, uc berkeley weyl character formula for compact lie groups.
I will now briefly describe a proof of the weyl kac character formula using lie algebra cohomology due to bott, kostant, garland and lepowsky 10, 14. In mathematics, the weyl character formula in representation theory describes the characters of irreducible representations of compact lie groups in terms of their highest weights. Random walks in weyl chambers and the decomposition of. Notes on weyl modules for semisimple algebraic groups. Bis a maximal torus contained in a borel subgroup of g. For convenience, we will discuss g, a reductive lie algebra over c, and a.
Character formula for the intergrable highest weight modules. Weyl character formula atiyahsegalsinger harishchandra character formula fixed point theorem p. Frenkels orbital theory for non twisted affine lie algebras to the case of twisted affine lie algebras using a character formula for certain nonconnected compact lie groups. A new character formula for lie algebras and lie groups dm. It is not, however, a formula that is combinatorial in nature, obscuring positivity. Weyl character formula in kktheory 3 dolbeault cohomology groups on bwith coe cients in e. Weyls character formula and lie algebra homology citeseerx. Jul 27, 20 the character formula of weyl 174 is the gem of the representation theory of compact lie groups. Weyl character formula let gbe a compact lie group with maximal torus t. Introduction to lie algebras and representation theory.
Let ru be the representation ring of u and be the weight lattice. The special linear lie algebra sl n,r consists of all n. It can also be seen as a quantization of the symmetric algebra \symv\, where \v\ is a finite dimensional vector space over a field of characteristic zero, by using a modified groenewold. Irreducible lie group characters and, more generally, spherical functions, are eigenfunctions of invariant differential operators, that. The weyl kac character formula gives a beautiful closedform expression for the characters of integrable highestweight modules of kacmoody algebras. However, unlike say the nite collection of symmetries of.
The proof of the weyl kac character formula for integrable highest weight modules now goes through much as in the. For compact lie groups, the peter weyl theorem, conjugacy of maximal tori two proofs, weyl character formula and more are covered. The authors compute the support varieties of all irreducible modules for the small quantum group u. Cahn, semisimple lie algebras and their representations. The weyl algebra is isomorphic to the quotient of the free algebra on two generators, x and y, by the ideal generated by the element. The weyl function and the kostant partition function march 22, 20 references. Classification and construction of semisimple lie algebras. Weyl s formula for the character of an irreducible representation, and its consequences the formula for the dimension of the representation, kostants. On the other hand, if g is the lie algebra of a lie group g, then there is an exponential map. Let b be a reductive lie subalgebra of a semisimple lie algebra f of the same rank both over the complex numbers. Even more important than the character formula itself is a corollary, the weyl dimension formula, which gives a fairly simple expression for the total dimension of the representation in terms of the root system. In this paper we interpret macdonalds unspecialized identities as multivariable vector partition theorems and we relate the wellknown rogersramanujan partition identities to the weyl kac character formula for an infinitedimensional euclidean generalized cartan matrix lie algebra.
Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. Harishchandras regularity theorem states that any invariant eigendistribution on a semisimple group or lie algebra is a locally integrable function. Weyls character formula for representations of semisimple. Even more important than the character formula itself is a corollary, the weyl dimension. In physics, a ne lie algebras are associated with conformal eld theories.
Let gbe a connected compact lie group with a maximal torus t. The short third part will be devoted to the proof weyl s character formula. B a borel subgroup n the unipotent radical of b t a maximal torus in b b the opposite of b. In this essay i shall present a proof of this formula that originated with kostant. Then the exponentiations on the right hand side of 1. We obtain a generalization of the weyl character formula. The weyl integral and character formulas we have seen that irreducible representations of a compact lie group g can be constructed starting from a highest weight space and applying negative roots to a highest weight vector. The special linear lie algebra sln,r consists of all n. Chevalley basis of the lie algebra a \divided power zform of its universal. The highest weight classi cation says that there is a onetoone correspondence between. The necessary background results on modules for sl 2c and complex semisimple lie algebras are outlined in the rst two sections.
I also have a mostly new chapter developing additional properties of the representations, including complete reducibility algebraic proof, the weyl character formula, the weyl dimension formula, and the kostant multiplicity formula. We can decompose the the diagrammatic antispherical hecke category in the following manner, h pnw m x2exp px y2exp py x. Further, we relate the weyl modules to the fusion modules defined in b. Weyl character formula for a connected and compact lie group. The compleat weyl character formula mark reeder november 28, 2014 1 notation and setup gis a connected reductive complex lie group t. We will discuss the classi cation of semisimple lie algebras, root systems, the weyl group, and dynkin diagrams. For compact lie groups, the book covers the peterweyl theorem, lie algebra, conjugacy of maximal tori, the weyl group, roots and weights, weyl character formula, the fundamental group and more. The addition and multiplication in the weyl character ring correspond to direct sum and tensor product of representations. The subalgebras of glv are referred to as the linear lie algebras. In this section well give this character formula and outline its proof. It is also a noncommutative example of a domain, and an example of an ore extension.
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