Harish-Chandra isomorphism

Let $\backslash mathfrak\{g\}$ be a semisimple Lie algebra, $\backslash mathfrak\{h\}$ its Cartan subalgebra and $\backslash lambda,\; \backslash mu\; \backslash in\; \backslash mathfrak\{h\}^*$ be two elements of the weight space (where $\backslash mathfrak\{h\}^*$ is the dual of $\backslash mathfrak\{h\}$) and assume that a set of positive roots $\backslash Phi\_+$ have been fixed. Let $V\_\backslash lambda$ and $V\_\backslash mu$ be highest weight modules with highest weights $\backslash lambda$ and $\backslash mu$ respectively.

The $\backslash mathfrak\{g\}$-modules $V\_\backslash lambda$ and $V\_\backslash mu$ are representations of the universal enveloping algebra $U(\backslash mathfrak\{g\})$ and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector). So, for $v\; \backslash in\; V\_\backslash lambda$ and $x\; \backslash in\; \backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$,

$$x\backslash cdot\; v:=\backslash chi\_\backslash lambda(x)v$$

and similarly for $V\_\backslash mu$, where the functions $\backslash chi\_\backslash lambda,\; \backslash ,\backslash chi\_\backslash mu$ are homomorphisms from $\backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$ to scalars called central characters.

For any $\backslash lambda,\; \backslash mu\; \backslash in\; \backslash mathfrak\{h\}^*$, the characters $\backslash chi\_\backslash lambda=\backslash chi\_\backslash mu$ if and only if $\backslash lambda\; +\; \backslash delta$ and $\backslash mu\; +\; \backslash delta$ are on the same orbit of the Weyl group of $\backslash mathfrak\{h\}^*$, where $\backslash delta$ is the half-sum of the positive roots, sometimes known as the Weyl vector.{{sfn|Humphreys|1978|page=130}}

Another closely related formulation is that the Harish-Chandra homomorphism from the center of the universal enveloping algebra $\backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$ to $S(\backslash mathfrak\{h\})^W$ (the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an isomorphism.

More explicitly, the isomorphism can be constructed as the composition of two maps, one from $\backslash mathfrak\{Z\}\; =\; \backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$ to $U(\backslash mathfrak\{h\})\; =\; S(\backslash mathfrak\{h\}),$ and another from $S(\backslash mathfrak\{h\})$ to itself.

The first is a projection $\backslash gamma:\; \backslash mathfrak\{Z\}\; \backslash rightarrow\; S(\backslash mathfrak\{h\})$. For a choice of positive roots $\backslash Phi\_+$, defining

$$n^+\; =\; \backslash bigoplus\_\{\backslash alpha\; \backslash in\; \backslash Phi\_+\}\; \backslash mathfrak\{g\}\_\backslash alpha,\; n^-\; =\; \backslash bigoplus\_\{\backslash alpha\; \backslash in\; \backslash Phi\_-\}\; \backslash mathfrak\{g\}\_\backslash alpha$$

as the corresponding positive nilpotent subalgebra and negative nilpotent subalgebra respectively, due to the Poincaré–Birkhoff–Witt theorem there is a decomposition

$$U(\backslash mathfrak\{g\})\; =\; U(\backslash mathfrak\{h\})\; \backslash oplus\; (U(\backslash mathfrak\{g\})\backslash mathfrak\{n\}^+\; +\; \backslash mathfrak\{n\}^-U(\backslash mathfrak\{g\})).$$

If $z\; \backslash in\; \backslash mathfrak\{Z\}$ is central, then in fact

$$z\; \backslash in\; U(\backslash mathfrak\{h\})\; \backslash oplus\; (U(\backslash mathfrak\{g\})\backslash mathfrak\{n\}^+\; \backslash cap\; \backslash mathfrak\{n\}^-U(\backslash mathfrak\{g\})).$$

The restriction of the projection $U(\backslash mathfrak\{g\})\; \backslash rightarrow\; U(\backslash mathfrak\{h\})$ to the centre is $\backslash gamma:\; \backslash mathfrak\{Z\}\; \backslash rightarrow\; S(\backslash mathfrak\{h\})$, and is a homomorphism of algebras. This is related to the central characters by

$$\backslash chi\_\backslash lambda(x)\; =\; \backslash gamma(x)(\backslash lambda)$$

The second map is the twist map $\backslash tau:\; S(\backslash mathfrak\{h\})\; \backslash rightarrow\; S(\backslash mathfrak\{h\})$. On $\backslash mathfrak\{h\}$ viewed as a subspace of $U(\backslash mathfrak\{h\})$ it is defined $\backslash tau(h)\; =\; h\; -\; \backslash delta(h)1$ with $\backslash delta$ the Weyl vector.

Then $\backslash tilde\; \backslash gamma\; =\; \backslash tau\; \backslash circ\; \backslash gamma:\; \backslash mathfrak\{Z\}\; \backslash rightarrow\; S(\backslash mathfrak\{h\})$ is the isomorphism. The reason this twist is introduced is that $\backslash chi\_\backslash lambda$ is not actually Weyl-invariant, but it can be proven that the twisted character $\backslash tilde\; \backslash chi\_\backslash lambda\; =\; \backslash chi\_\{\backslash lambda\; -\; \backslash delta\}$ is.

The theorem has been used to obtain a simple Lie algebraic proof of Weyl's character formula for finite-dimensional irreducible representations.{{sfn|Humphreys|1978|pages=135–141}} The proof has been further simplified by Victor Kac, so that only the quadratic Casimir operator is required; there is a corresponding streamlined treatment proof of the character formula in the second edition of {{harvtxt|Humphreys|1978|pages=143–144}}.

Further, it is a necessary condition for the existence of a non-zero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for Verma modules or generalized Verma modules $V\_\backslash lambda$ with highest weight $\backslash lambda$, there exist only finitely many weights $\backslash mu$ for which a non-zero homomorphism $V\_\backslash lambda\; \backslash rightarrow\; V\_\backslash mu$ exists.

For $\backslash mathfrak\{g\}$ a simple Lie algebra, let $r$ be its rank, that is, the dimension of any Cartan subalgebra $\backslash mathfrak\{h\}$ of $\backslash mathfrak\{g\}$. H. S. M. Coxeter observed that $S(\backslash mathfrak\{h\})^W$ is isomorphic to a polynomial algebra in $r$ variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore, the center of the universal enveloping algebra of a simple Lie algebra is isomorphic to a polynomial algebra. The degrees of the generators of the algebra are the degrees of the fundamental invariants given in the following table.

The number of the fundamental invariants of a Lie group is equal to its rank. Fundamental invariants are also related to the cohomology ring of a Lie group. In particular, if the fundamental invariants have degrees $d\_1,\; \backslash cdots,\; d\_r$, then the generators of the cohomology ring have degrees $2d\_1-1,\; \backslash cdots,\; 2d\_r-1$. Due to this, the degrees of the fundamental invariants can be calculated from the Betti numbers of the Lie group and vice versa. In another direction, fundamental invariants are related to cohomology of the classifying space. The cohomology ring $H^*(BG,\; \backslash mathbb\{R\})$ is isomorphic to a polynomial algebra on generators with degrees $2d\_1,\; \backslash cdots,\; 2d\_r$.{{cite journal |last1=Borel |first1=Armand |title=Sur la cohomologie des espaces homogenes des groupes de Lie compacts |journal=American Journal of Mathematics |date= Apr 1954 |volume=76| issue = 2| pages= 273–342 }}

• If $\backslash mathfrak\{g\}$ is the Lie algebra $\backslash mathfrak\{sl\}(2,\; \backslash mathbb\{R\})$, then the center of the universal enveloping algebra is generated by the Casimir invariant of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to $\backslash mathbb\{R\}$, by negation, so the invariant of the Weyl group is the square of the generator of the Cartan subalgebra, which is also of degree 2.
• For $\backslash mathfrak\{g\}\; =\; A\_2\; =\; \backslash mathfrak\{sl\}(3,\backslash mathbb\{C\})$, the Harish-Chandra isomorphism says $\backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$ is isomorphic to a polynomial algebra of Weyl-invariant polynomials in two variables $h\_1,\; h\_2$ (since the Cartan subalgebra is two-dimensional). For $A\_2$, the Weyl group is $S\_3\; \backslash cong\; D\_6$ which acts on the CSA in the standard representation. Since the Weyl group acts by reflections, they are isometries and so the degree 2 polynomial $f\_2(h\_1,\; h\_2)\; =\; h\_1^2\; +\; h\_2^2$ is Weyl-invariant. The contours of the degree 3 Weyl-invariant polynomial (for a particular choice of standard representation where one of the reflections is across the x-axis) are shown below. These two polynomials generate the polynomial algebra, and are the fundamental invariants for $A\_2$.
• For all the Lie algebras in the classification, there is a fundamental invariant of degree 2, the quadratic Casimir. In the isomorphism, these correspond to a degree 2 polynomial on the CSA. Since the Weyl group acts by reflections on the CSA, they are isometries, so the degree 2 invariant polynomial is $f\_2(\backslash mathbf\{h\})\; =\; h\_1^2\; +\; \backslash cdots\; +\; h\_r^2$ where $r$ is the dimension of the CSA $\backslash mathfrak\{h\}$, also known as the rank of the Lie algebra.
• For $\backslash mathfrak\{g\}\; =\; A\_1\; =\; \backslash mathfrak\{sl\}(2,\; \backslash mathbb\{C\})$, the Cartan subalgebra is one-dimensional, and the Harish-Chandra isomorphism says $\backslash mathcal\{Z\}(U(\backslash mathfrak\{g\}))$ is isomorphic to the algebra of Weyl-invariant polynomials in a single variable $h$. The Weyl group is $S\_2$ acting as reflection, with non-trivial element acting on polynomials by $h\; \backslash mapsto\; -h$. The subalgebra of Weyl-invariant polynomials in the full polynomial algebra $K[h]$ is therefore only the even polynomials, generated by $f\_2(h)\; =\; h^2$.

{{multiple image

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• For $\backslash mathfrak\{g\}\; =\; B\_2\; =\; \backslash mathfrak\{so\}(5)\; =\; \backslash mathfrak\{sp\}(4)$, the Weyl group is $D\_8$, acting on two coordinates $h\_1,\; h\_2$, and is generated (non-minimally) by four reflections, which act on coordinates as $(h\_1\; \backslash mapsto\; -h\_1,\; h\_2\; \backslash mapsto\; h\_2),\; (h\_1\; \backslash mapsto\; h\_1,\; h\_2\; \backslash mapsto\; -h\_2),\; (h\_1\; \backslash mapsto\; h\_2,\; h\_2\; \backslash mapsto\; h\_1),\; (h\_1\; \backslash mapsto\; -h\_2,\; h\_2\; \backslash mapsto\; h\_1)$. Any invariant quartic must be even in both $h\_1$ and $h\_2$, and invariance under exchange of coordinates means any invariant quartic can be written $f\_4(h\_1,\; h\_2)\; =\; ah\_1^4\; +\; bh\_1^2h\_2^2\; +\; ah\_2^4.$ Despite this being a two-dimensional vector space, this contributes only one new fundamental invariant as $f\_2(h\_1,h\_2)^2$ lies in the space. In this case, there is no unique choice of quartic invariant as any polynomial with $b\; \backslash neq\; 2a$ (and $a,b$ not both zero) suffices.

The above result holds for reductive, and in particular semisimple Lie algebras. There is a generalization to affine Lie algebras shown by Feigin and Frenkel showing that an algebra known as the Feigin–Frenkel center is isomorphic to a W-algebra associated to the Langlands dual Lie algebra $^L\backslash mathfrak\{g\}$.{{cite journal |last1=Molev |first1=Alexander |title=On Segal–Sugawara vectors and Casimir elements for classical Lie algebras |journal=Letters in Mathematical Physics |date=19 January 2021 |volume=111 |issue=8 |doi=10.1007/s11005-020-01344-3 |arxiv=2008.05256 |s2cid=254795180 }}{{cite journal |last1=Feigin |first1=Boris |last2=Frenkel |first2=Edward |last3=Reshetikhin |first3=Nikolai |title=Gaudin Model, Bethe Ansatz and Critical Level |journal=Commun. Math. Phys. |date=3 Apr 1994 |volume=166 |pages=27–62 |doi=10.1007/BF02099300 |arxiv=hep-th/9402022 |s2cid=17099900 }}

The Feigin–Frenkel center of an affine Lie algebra $\backslash hat\; \backslash mathfrak\{g\}$ is not exactly the center of the universal enveloping algebra $\backslash mathcal\{Z\}(U(\backslash hat\; \backslash mathfrak\{g\}))$. They are elements $S$ of the vacuum affine vertex algebra at critical level $k\; =\; -h^\backslash vee$, where $h^\backslash vee$ is the dual Coxeter number for $\backslash mathfrak\{g\}$ which are annihilated by the positive loop algebra $\backslash mathfrak\{g\}[t]$ part of $\backslash hat\; \backslash mathfrak\{g\}$, that is,

$$\backslash mathfrak\{Z\}(\backslash hat\; \backslash mathfrak\{g\})\; :=\; \backslash \{S\; \backslash in\; V\_\{\backslash text\{cri\}\}(\backslash mathfrak\{g\})\; |\; \backslash mathfrak\{g\}[t]S\; =\; 0\backslash \}$$

where $V\_\{\backslash text\{cri\}\}(\backslash mathfrak\{g\})$ is the affine vertex algebra at the critical level. Elements of this center are also known as singular vectors or Segal–Sugawara vectors.

The isomorphism in this case is an isomorphism between the Feigin–Frenkel center and the W-algebra constructed associated to the Langlands dual Lie algebra by Drinfeld–Sokolov reduction:

$$\backslash mathfrak\{Z\}(\backslash hat\; \backslash mathfrak\{g\})\; \backslash cong\; \backslash mathcal\{W\}(^L\backslash mathfrak\{g\}).$$

There is also a description of $\backslash mathfrak\{Z\}(\backslash hat\; \backslash mathfrak\{g\})$ as a polynomial algebra in a finite number of countably infinite families of generators, $\backslash partial^n\; S\_i,\; i\; =\; 1,\; \backslash cdots,\; l,\; n\; \backslash geq\; 0$, where $S\_i,\; i\; =\; 1,\; \backslash cdots,\; l$ have degrees $d\_i\; +1,\; i\; =\; 1,\; \backslash cdots,\; l$ and $\backslash partial$ is the (negative of) the natural derivative operator on the loop algebra.

• Translation functor
• Infinitesimal character

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[https://www2.math.upenn.edu/~brweber/Courses/2013/Math651/Notes/L14_HarishChandra.pdf Notes on the Harish-Chandra isomorphism]

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• {{Citation | last1=Harish-Chandra | title=On some applications of the universal enveloping algebra of a semisimple Lie algebra | jstor=1990524 | mr=0044515 | year=1951 | journal=Transactions of the American Mathematical Society | volume=70 | issue=1 | pages=28–96 | doi=10.2307/1990524| doi-access=free }}
• {{cite book | mr = 0499562 | last= Humphreys| first =James E.| title=Introduction to Lie algebras and representation theory|

edition= Second revised| series= Graduate Texts in Mathematics| volume = 9| publisher= Springer-Verlag | year= 1978| isbn = 0-387-90053-5}} (Contains an improved proof of Weyl's character formula.)

• {{Citation | last1=Humphreys | first1=James E. | author1-link=James E. Humphreys | title=Representations of semisimple Lie algebras in the BGG category O | publisher=AMS | year=2008 | isbn=978-0-8218-4678-0 | page=26}}
• {{Citation | last1=Knapp | first1=Anthony W. | first2=David A. |last2=Vogan| title=Cohomological induction and unitary representations | publisher=Princeton University Press | series=Princeton Mathematical Series | isbn=978-0-691-03756-1 | mr=1330919 | year=1995 | volume=45}}
• {{citation |first1=Anthony W. |last1=Knapp |author1-link=Anthony W. Knapp |chapter=V. Finite Dimensional Representations §5. Harish-Chandra Isomorphism |title=Lie Groups Beyond an Introduction |chapter-url=https://books.google.com/books?id=J8EGCAAAQBAJ&pg=PA246 |date=2013 |orig-year=1996 |publisher=Springer |isbn=978-1-4757-2453-0 |pages=246–258 |volume=140 |series=Progress in Mathematics}}

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Category:Representation theory of Lie algebras

Category:Theorems in algebra