versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.1 Bahía Blanca ene./jun. 2008
Abstract. The -homology groups of a -module provide a natural and fruitful extension of the concept of highest weight to the representation theory of a noncompact reductive Lie group. In this article we give an introduction to the -homology groups and a survey of some developments, with a particular emphasis on results pertaining to the problem of caculating -homology groups.
The concept of a highest weight and its use to classify irreducible representations of compact Lie groups can traced back nearly a century, to seminal work by E. Cartan and H. Weyl. For a compact, connected Lie group, the highest weight theory gives a tight parametrization of irreducible representations in terms of specific invariants associated to the group. If one tries to extend this concept to the representation theory of a noncompact, real reductive group one immediately encounters two problems. On the one hand, in the noncompact case, it turns out there are several conjugacy classes of complex Borel subalgebras, and what might be called a highest weight depends on the choice of a conjugacy class. On the other hand, it is quite common that what should be called a highest weight turns out to be zero for every choice of Borel subalgebra. This means, in the traditional sense, the highest weight does not exist for a great majority of irreducible representations.
Although there is no way to avoid the first problem, representation theorists have confronted the second problem by considering the highest weight to be a functorial construction and studying the related derived functors. This has proved to be especially fruitful, producing a strong and useful family of invariants associated to a representation. In this article we give a brief introduction to the -homology (and -cohomology) groups, followed by a survey of some results, focusing on developments related to the problem of calculating the -homology of representations.
The author would like to take this opportunity to thank the organizers of the 2007 meeting of the UMA for giving him an opportunity to present some results in the form of a conference and for asking him to submit this article. He would also like to acknowledge the help and encouragement he has received from Jorge Vargas. This article is dedicated to the memory of Mischa Cotlar, with a special recognition to Dr. Cotlar's role as advisor and mentor to the late José Pererya.
In this section we introduce the -homology and -cohomology of -modules (for more details see ).
Let be a complex reductive Lie algebra. By definition, a Borel subalgebra of is a maximal solvable subalgebra and a parabolic subalgebra of is a subalgebra that contains a Borel subalgebra. If is a parabolic subalgebra then the nilradical of is the largest solvable ideal in . A Levi factor is a complementary subalgebra to in . One knows that Levi factors exist and that they are exactly the subalgebras which are maximal with respect to being reductive in . When is a Levi factor than
is called a Levi decomposition.
Fix a parabolic subalgebra with nilradical and Levi factor . Let denote the enveloping algebra of and let be the 1-dimensional trivial module. If is a -module then the zero homology of is the -module
This -module is sometimes referred to as the space of coinvariants, although it clearly depends on the choice of parabolic subalgebra. The definition of the zero homology determines a right exact functor from the category of -modules to the category of -modules. The -homology groups of are the -modules obtained as the corresponding derived functors. There is a standard complex for calculating these homology groups, defined as follows. The right standard resolution of is the complex of free right -modules given by
Applying the functor
to the standard resolution we obtain a complex
of left -modules called the standard -homology complex. Here acts via the tensor product of the adjoint action on with the given action on . Since is a free -module, a routine homological argument identifies the pth homology of the standard complex with the pth -homology group
One can prove that the induced -action on the homology groups of the standard complex is the correct one.
The zero -cohomology of a -module is the -module
This -module is sometimes referred to as the space of invariants, and also clearly depends on the choice of parabolic subalgebra The definition of the zero cohomology determines a left exact functor from the category of -modules to the category of -modules. By definition, the -cohomology groups of are the -modules obtained as the corresponding derived functors. These -modules can be calculated by applying the functor
to the standard resolution of , this time by free left -modules. In a natural way, one obtains a complex of -modules and the pth cohomology of this complex realizes the pth -cohomology group
It turns out that the structure of the -cohomology is determined by the structure of the -homology, in a simple way . Thus, it is often a matter of convenience whether one works with homology groups or cohomology groups. In this article, we will focus on results framed in terms of homology. The following proposition, whose proof is established by an analysis of standard complexes, can be used to translate results about -homology into results about -cohomology [9, Section 2].
In this section we review some classical results about the representation theory of reductive Lie groups (for details see ) and introduce the canonical globalizations.
For simplicity we work with a class of reductive Lie groups we call linear, although there is no problem working in the more general context of a reductive group of Harish-Chandra class. In particular, we assume the following setup. will denote a connected, complex reductive group. This means is a connected, complex Lie group with the property that the maximal compact subgroups are real forms of . The group will denote a real form of and is assumed to have finitely many connected components. We call a linear reductive Lie group. The Lie algebras of and will be denoted and , respectively. For the remainder of this article we fix a maximal compact subgroup of and let be the complexification of . In general, we write etc. to indicate complex subgroups of and denote the corresponding Lie algebras by etc. Subgroups of will be denoted by etc. with the corresponding real Lie algebras written as etc.
A representation of will mean a continuous linear action of in a complete, locally convex topological vector space. When we speak of irreducible or finite length representations, the corresponding definitions should be framed in terms of invariant closed subspaces. A vector in a representation is called smooth when
In order to define -homology groups, we will be primarily interested in smooth representations. These are representations where every vector is smooth. In a natural way a smooth representation carries a compatible -action. For a compact Lie group one can show that a finite length representation is finite-dimensional and therefore smooth.
We recall some basic results about the infinite-dimensional representations of reductive groups. In the 1950s, Harish-Chandra proved that an irreducible unitary representation has the property that each irreducible -submodule has finite multiplicity in . This led him to define and study admissible representations. By definition, this means each irreducible -submodule has finite multiplicity. Harish-Chandra then considered the subspace of -finite vectors. By definition, a vector in a representation is called -finite if the span of the -orbit of is finite-dimensional. Although the subspace of -finite vectors is not -invariant, Harish-Chandra proved that -finite vectors are smooth, and thus form a -module called the underlying Harish-Chandra module.
On the other hand, it is possible to define abstractly the concept of a Harish-Chandra module. This is a -module equipped with a compatible, locally finite -action. Harish-Chandra proved that an irreducible Harish-Chandra module appears as the underlying -module of -finite vectors in an irreducible admissible Banach space representation for and W. Casselman proved that the same holds for any finite-length Harish-Chandra module. By now we know more. In particular, given a Harish-Chandra module we define a globalization of to be an admissible representation for whose underlying Harish-Chandra is . We assume our Harish-Chandra modules have finite-length. Then we can assert that several canonical and functorial globalizations exist on the category of Harish-Chandra modules. These are: the smooth globalization of Casselman and Wallach , its dual (called: the distribution globalization), Schmid's minimal globalization  and its dual (the maximal globalization). All four globalizations are smooth. We will let , , and denote respectively, the minimal, the maximal, the smooth and the distribution globalizations of a Harish-Chandra module . If denotes a Banach globalization of , then there is a natural chain of inclusions
In this chain the minimal globalization is known to coincide with the analytic vectors in while coincides with the smooth vectors in . In particular, one knows that a finite-length admissible Banach space representation for is smooth if and only if it is finite dimensional. Later in this article we will review various results, often called comparison theorems, relating the -homologies of a Harish-Chandra module to the -homologies of a canonical globalization.
In this section we recall some structure theory and an important technical result about the decomposition of -homology groups for certain -modules, giving special emphasis on the case of a Borel subalgebra. Recall that a Cartan subalgebra is a maximal abelian subalgebra whose elements are semisimple under the adjoint representation of in . A nonzero eigenvalue for the adjoint representation is called a root. will denote the set of roots. Thus
where is the eigenspace corresponding to root . One knows that if and only if . If is a Borel subalgebra of containing then the roots of in define a subset called the corresponding set of positive roots. When the sum of two positive roots is a root, then that sum is positive. One also knows that is a disjoint union:
One can show there is a unique such that We use this element to define the value of the dual root. In particular, the dual root is given by
The linear reflection corresponding to is defined as
These reflections generate a finite subgroup of the general linear group of , denoted and called the Weyl group of in .
Let denote the center of the enveloping algebra of . A -infinitesimal character is a homomorphism of algebras
Since acts on an irreducible Harish-Chandra module (and also any corresponding smooth globalization) by a scalar, the infinitesimal character is an important invariant associated to an irreducible, admissible representation. We now recall Harish-Chandra's parametrization of infinitesimal characters. We choose a Borel subalgebra containing . Thus
Then one knows that and that the corresponding projection of in defines an injective morphism of algebras called the unnormalized Harish-Chandra map. We can use this morphism to identify infinitesimal characters with Weyl group orbits in in the following way. Let denote one-half the sum of the positive roots and suppose . Then, via the unnormalized Harish-Chandra map, the composition
defines an infinitesimal character . One knows that for , the element defines the same infinitesimal character . Abusing notation somewhat, we write . The infinitesimal character is called regular when the only element of fixing an element in the orbit , is the identity. This is equivalent to the condition that
For a -module with regular infinitesimal character one has the following result. The notes by D. Milicic  contain a proof.
Let such that and let be one half the sum of the positive roots. Then the Cartan subalgebra acts semisimply on the -homology groups with eigenvalues of the form for . In particular
A generalization of this result works for any parabolic subalgebra with Levi decomposition
although the parametrization is not quite as tight. In particular, if is a -module with regular infinitesimal character and if denotes the center of the enveloping algebra of then is a semisimple -module and decomposes into a direct sum of -eigenspaces, where the associated -infinitesimal characters that appear are related to by an appropriately defined Harish-Chandra map.
When is a connected, compact Lie group, there is a result, called Kostant's theorem, that calculates the -homolgy groups of an irreducible representation. In this section we review that result, with special emphasis on the case of a Borel subalgebra.
Assume is a compact real form of . Fix a Borel subalgebra . Then the normalizer of in is a maximal torus and a real form for a Cartan subgroup of . We let be the Lie algebra of and the nilradical of . is the set of roots. The roots of in determine a set of positive roots . Let be one-half the sum of the positive roots. Suppose
is a continuous character and let denote the complexification of the derivative of . To be consistent with the notation in Section 7 we use the shifted parameter
One knows that
The character is called antidominant and regular if
The Cartan-Weyl parametrization of irreducible representations is as follows.
Theorem 5.1. Maintain the established notations.
(a) Suppose is an irreducible -module. Then the space of coinvariants is an irreducible -module and the associated character is antidominant and regular. This character is called the lowest weight.
(b) If two irreducible representations have the same lowest weight then they are isomorphic.
(c) To each antidominant and regular character there is an irreducible -module with the given character as its lowest weight.
We need to define the length function on the Weyl group. One knows that the Weyl group permutes the roots of in . We can define the length of to be the number of roots in
Kostant's theorem is the following:
Theorem 5.2. Suppose is the irreducible representation for with lowest weight and let be the shifted parameter. Then is a sum of irreducible -modules each having multiplicity one. The characters of that show up as eigenvalues in are exactly those whose derivative have the form where the length of is p
In the more general case of a parabolic subalgebra , let be the normalizer of in and let be the complexified Lie algebra of . One knows that is connected and that is a Levi factor of . Indeed, if is the connected subgroup of with Lie algebra then is the compact real form of . Suppose is an irreducible representation for and let be the nilradical of . Then Kostant's Theorem describes the structure of the pth homology group as an -module. In particular, the theorem states that an irreducible representation of has, at most, multiplicity one in and gives a precise condition when appears, in terms of the degree p, the lowest weight of and the lowest weight of . We refer the reader to [10, Chapter IV, Section 9] for more details.
As we mentioned before, when is noncompact, there are several conjugacy classes of Borel subalgebras and the structure of the -homology groups of a representation can depend on the choice of -conjugacy class. On the other hand, when is a Harish-Chandra module, then the locally finite -action on extends naturally to a locally holomorphic -action, and it turns out that the -homology groups of depend on the -conjugacy classes of Borel subalgebras. In order to compare the -homology groups of with the -homology groups of a smooth globalization, we therefore need to know something about the relationship between -conjugacy classes and -conjugacy classes. There is an elegant geometric result, referred to as Matsuki duality, that gives us the needed information. We now review that result.
One knows that the group acts transitively on the set of Borel subalgebras of . The corresponding -homogeneous complex manifold is called the full flag space. In general, if is a parabolic subalgebra of then the normalizer of in is the connected subgroup with Lie algebra and the corresponding quotient
is called a flag manifold. The points in are naturally identified with the -conjugates to .
Let be the complexification of a Cartan involution of corresponding to the maximal compact subgroup . A Cartan subalgebra of is called stable if is a real form and if . A Borel subalgebra is called very special if it contains a stable Cartan subalgebra. A stable Cartan subalgebra of a Borel subalgebra is unique. A point in the full flag space is called very special if the corresponding Borel subalgebra is.
Matsuki has established the following .
It follows that the very special points give a one-to-one correspondence between the -orbits and the -orbits on , defined by the following duality. A -orbit is said to be dual to a -orbit when contains a special point. In this duality, open -orbits correspond to closed -orbits and the (unique) closed -orbit corresponds to the (unique) open -orbit. We note that Matsuki has established a similar result for any flag manifold .
Example 6.2. Suppose , the group of complex matrices with determinant 1 and let . Then the full flag space is isomorphic to the Riemann sphere. has three orbits on . The closed -orbit can be identified with an equatorial circle and the other two orbits are the corresponding open hemispheres. It turns out every point in the closed orbit is very special, independent of the choice of (this is true in general). Put . Thus . Then the three -orbits on are a punctured plane, containing the closed -orbit, and two fixed points, which can be identified with the respective poles in each of the open hemispheres. These two poles are the other very special points.
When is a Harish-Chandra module and is the nilradical of a Borel subalgebra then one knows that the homology groups are finite-dimensional, so it may seem reasonable to ask when coincides with the -homolgy groups of a smooth globalization. It turns out this not only depends on the choice of Borel subalgebra, but also in the the choice of smooth globalization. When is the nilradical of a very special Borel subalgebra, is a Harish-Chandra module, and is the minimal globalization, then H. Hecht and J. Taylor have shown  that the natural map
On the other hand, for the maximal globalization, there are counterexamples to this result.
The result of Hecht and Taylor has been generalized in the following form. A Levi factor of a parabolic subalgebra is called stable if is a real form of and if . The parabolic subalgebra is called very special if it contains a stable Levi factor. Such a Levi factor is unique. Unlike the case of the full flag space, there may be parabolic subalgebras which are not -conjugate to a very special parabolic subalgebra, so we are not considering all orbits on every flag manifold. However, suppose is very special and is the stable Levi factor. Define to be the subgroup of that normalizes and normalizes . Then is a linear reductive Lie group with complexified Lie algebra and maximal compact subgroup , called the associated real Levi subgroup. We have have the following result [9, Proposition 2.24].
Proposition 6.3. Suppose is a very special parabolic subalgebra with and defined as above. Let be the nilradical of and suppose is a Harish-Chandra module for . Then the -homology groups are Harish-Chandra modules for ,.
For the minimal globalization, we have the following .
One might conjecture that above theorem works for the smooth globalization, and W. Casselman has informed the author that he has proven something along these lines, although details are unclear. Two partial comparison theorems about smooth globalizations have been published by other mathematicians. H. Hecht and J. Taylor have shown the result for minimal parabolic subgroups of , while U. Bunke and M. Olbrich have shown the result for any real parabolic subgroup .
D. Vogan has conjectured that all four canonical globalizations commute with the -homology groups of a very special parabolic subalgebra when the corresponding -orbit on the flag manifold is open . We remark that it has recently been shown that Vogan's conjecture is true for one globalization if and only it's true for the dual . Thus the conjecture is proven for both the minimal and maximal globalization.
In the noncompact case, the problem of calculating -homology groups can be quite complicated and there seems to be little hope of just writing down a formula that generalizes Kostant's theorem for all irreducible representations. However, there are certain representations, called standard modules, whose -homology groups are a bit more predictable. These standard modules are generically irreducible, coincide with irreducibles when is compact, and can be used to classify the irreducible representations. In this section we define the standard representations and consider their -homology groups, focusing on the case of the full flag space.
In particular, we use the construction of minimal globalizations given in . Let be the full flag space and, since we need to keep track of points, introduce the following notation. For we let be the corresponding Borel subalgebra and let denote the nilradical of . When we are interested in calculating the -homology of Harish-Chandra modules, we can assume is a very special Borel subalgebra. In that case, denotes the stable Cartan subalgebra of and is the corresponding real Cartan subgroup (thus is the associated Levi subgroup). By our linear assumptions on , it follows that is abelian, so that an irreducible, admissible representation of is a continuous character
Let be the -orbit of . In a natural way, extends to a character of the normalizer of in (we note that and the normalizer of coincide exactly when is open). Thus determines a -homogeneous analytic line bundle over . One can then define the concept of a polarized section [7, Section 8]. When is open, the polarized sections are holomorphic sections, and in general the polarized sections are locally isomorphic with the restricted holomorphic functions. Let denote the sheaf of polarized sections on and, for p let
denote the corresponding compactly supported sheaf cohomology group. Suppose is the complexified differential of , let be one-half the sum of the positive roots for in and let denote the corresponding shifted parameter. Thus
We have the following theorem .
Theorem 7.1. Maintain the previously defined notations. Let q be the codimension of the -orbit of in .
(a) carries a natural topology and a continuous -action, so that the resulting representation is a minimal globalization.
(b) has infinitesimal character .
(c) When is antidominant then when pq.
(d) When is antidominant and is regular then contains a unique irreducible submodule. In particular, .
When is antidominant and is regular, we call a regular standard module. These modules can be used to parametrize irreducible representations with regular infinitesimal character. For the remainder of this article we will make some remarks about how to calculate the -homology of regular standard modules. But we first note that, in the case of a singular infinitesimal character, the definition of standard module is more subtle, and the calculation of -homology is more elusive.
To state results we will need to differentiate points where we calculate -homology and the corresponding parameters for eigenvalues of a Cartan subalgebra (see Theorem 4.1). In particular, we fix a very special point as a base point. For we put . When is a very special Borel subalgebra and is the stable Cartan subalgebra, then there exists such that
This isomorphism is independent of the choice of . For put
We note that is a root of in is a root of and that is positive at is positive at . In particular, is one-half the sum of the positive roots for in .
The circle of ideas utilized in  depend on an identification of the derived functor of -homolgy, in a certain weight (Theorem 4.1), with the geometric fiber applied to a certain, corresponding localization functor. These ideas originate in the an elegant generalization of Casselman's submodule theorem, given by A. Beilinson and J. Bernstein in . This identification, together with some functorial rigmarole, immediately leads to the following result.
Proposition 7.2. Suppose is a regular standard module. Maintain the previously introduced notations. Let denote the 1-dimensional representation of corresponding to and let be the nilradical of . Then we have the following.
(b) If is a very special Borel subalgebra with nilradical and then
According to Theorem 4.1, the problem of calculating of the -homology groups of , at a special point , reduces to the problem of calculating the values in the weights for . In geometric terms, this means calculating the geometric fibers of certain localizations of or, equivalently, calculating the result of the so called intertwining functor. We briefly consider this problem.
In general, a positive root is called simple if it cannot be decomposed into a nontrivial sum of positive roots. Let be the positive roots associated to in . For a simple root , the problem of calculating the values of the -homology groups in the weight can be geometrically reduced to specific calculations for certain real subgroups of . To a large extent, this idea is already exploited and explained in  and some of the necessary calculations are dealt with there.
We finish with an example where, using these ideas, a general formula, like Kostant's, can be actually written down. Assume is a connected, complex reductive Lie group. Fix a very special Borel subalgebra with stable Cartan subalgebra and let denote the corresponding positive roots. For each , the set
defines a new set of positive roots and thus a corresponding Borel subalgebra of containing the stable Cartan subalgebra . Thus the point is very special. Because is a complex reductive group, one knows that each Borel subalgebra of is -conjugate to a Borel subalgebra of the form . Suppose is the Cartan subgroup of with Lie algebra . Then each defines a holomorphic character of and by restriction, a corresponding character of . We write for this character of . If we let be the shifted parameter and put then .
Using the above ideas, one can deduce the following.
Theorem 7.3. Let be a connected, complex reductive group. Suppose is the previously defined regular standard module and assume the -orbit of is open in . We define a chain of simple roots to be a finite sequence of roots of such that for each , is simple for the set of positive roots defined by
Suppose is a chain of simple roots and let be the ordered product of reflections given by this chain. Let be the character of defined by
and let be the 1-dimensional representation of corresponding to the character . Let q denote the codimension of the -orbit of in . Then
We note that the hypothesis of the theorem implies that the representation is irreducible, and also remark that any attempt to write down a similar result for other orbits (even in the case of a connected, complex reductive group), when the standard module is reducible, is considerably more complicated.
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Facultad de Ciencias Exactas,
UNICEN. Tandil, Argentina
Recibido: 10 de abril de 2008
Aceptado: 6 de mayo de 2008