Verma module

Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Specifically, although Verma modules themselves are infinite dimensional, quotients of them can be used to construct finite-dimensional representations with highest weight formula_1, where formula_1 is dominant and integral. Their homomorphisms correspond to invariant differential operators over flag manifolds.
Informal construction.
We can explain the idea of a Verma module as follows. Let formula_3 be a semisimple Lie algebra (over formula_4, for simplicity). Let formula_5 be a fixed Cartan subalgebra of formula_3 and let formula_7 be the associated root system. Let formula_8 be a fixed set of positive roots. For each formula_9, choose a nonzero element formula_10 for the corresponding root space formula_11 and a nonzero element formula_12 in the root space formula_13. We think of the formula_10's as "raising operators" and the formula_12's as "lowering operators."
Now let formula_16 be an arbitrary linear functional, not necessarily dominant or integral. Our goal is to construct a representation formula_17 of formula_3 with highest weight formula_1 that is generated by a single nonzero vector formula_20 with weight formula_1. The Verma module is one particular such highest-weight module, one that is maximal in the sense that every other highest-weight module with highest weight formula_1 is a quotient of the Verma module. It will turn out that Verma modules are always infinite dimensional; if formula_1 is dominant integral, however, one can construct a finite-dimensional quotient module of the Verma module. Thus, Verma modules play an important role in the classification of finite-dimensional representations of formula_3. Specifically, they are an important tool in the hard part of the theorem of the highest weight, namely showing that every dominant integral element actually arises as the highest weight of a finite-dimensional irreducible representation of formula_3.
We now attempt to understand intuitively what the Verma module with highest weight formula_1 should look like. Since formula_20 is to be a highest weight vector with weight formula_1, we certainly want 
and
Then formula_17 should be spanned elements obtained by lowering formula_20 by the action of the formula_12's:
We now impose "only" those relations among vectors of the above form required by the commutation relations among the formula_35's. In particular, the Verma module is always infinite-dimensional. The weights of the Verma module with highest weight formula_1 will consist of all elements formula_37 that can be obtained from formula_1 by subtracting integer combinations of positive roots. The figure shows the weights of a Verma module for formula_39.
A simple re-ordering argument shows that there is only one possible way the full Lie algebra formula_3 can act on this space. Specifically, if formula_41 is any element of formula_3, then by the easy part of the Poincaré–Birkhoff–Witt theorem, we can rewrite
as a linear combination of products of Lie algebra elements with the raising operators formula_10 acting first, the elements of the Cartan subalgebra, and last the lowering operators formula_12. Applying this sum of terms to formula_20, any term with a raising operator is zero, any factors in the Cartan act as scalars, and thus we end up with an element of the original form.
To understand the structure of the Verma module a bit better, we may choose an ordering of the positive roots as formula_47 and we let the corresponding lowering operators by formula_48. Then by a simple re-ordering argument, every element of the above form can be rewritten as a linear combination of elements with the formula_35's in a specific order:
where the formula_51's are non-negative integers. Actually, it turns out that such vectors form a basis for the Verma module.
Although this description of the Verma module gives an intuitive idea of what formula_17 looks like, it still remains to give a rigorous construction of it. In any case, the Verma module gives—for "any" formula_1, not necessarily dominant or integral—a representation with highest weight formula_1. The price we pay for this relatively simple construction is that formula_17 is always infinite dimensional. In the case where formula_1 is dominant and integral, one can construct a finite-dimensional, irreducible quotient of the Verma module.
The case of sl(2; C).
Let formula_57 be the usual basis for formula_58:
with the Cartan subalgebra being the span of formula_60. Let formula_1 be defined by formula_62 for an arbitrary complex number formula_63. Then the Verma module with highest weight formula_1 is spanned by linearly independent vectors formula_65 and the action of the basis elements is as follows:
(This means in particular that formula_67 and that formula_68.) These formulas are motivated by the way the basis elements act in the finite-dimensional representations of formula_58, except that we no longer require that the "chain" of eigenvectors for formula_60 has to terminate.
In this construction, formula_63 is an arbitrary complex number, not necessarily real or positive or an integer. Nevertheless, the case where formula_63 is a non-negative integer is special. In that case, the span of the vectors formula_73 is easily seen to be invariant—because formula_74. The quotient module is then the finite-dimensional irreducible representation of formula_58 of dimension formula_76
Definition of Verma modules.
There are two standard constructions of the Verma module, both of which involve the concept of universal enveloping algebra. We continue the notation of the previous section: formula_3 is a complex semisimple Lie algebra, formula_5 is a fixed Cartan subalgebra, formula_7 is the associated root system with a fixed set formula_8 of positive roots. For each formula_9, we choose nonzero elements formula_82 and formula_83.
As a quotient of the enveloping algebra.
The first construction of the Verma module is a quotient of the universal enveloping algebra formula_84 of formula_3. Since the Verma module is supposed to be a formula_3-module, it will also be a formula_84-module, by the universal property of the enveloping algebra. Thus, if we have a Verma module formula_17 with highest weight vector formula_20, there will be a linear map formula_90 from formula_84 into formula_17 given by
Since formula_17 is supposed to be generated by formula_20, the map formula_90 should be surjective. Since formula_20 is supposed to be a highest weight vector, the kernel of formula_90 should include all the root vectors formula_10 for formula_100 in formula_8. Since, also, formula_20 is supposed to be a weight vector with weight formula_1, the kernel of formula_90 should include all vectors of the form
Finally, the kernel of formula_90 should be a left ideal in formula_84; after all, if formula_108 then formula_109 for all formula_110.
The previous discussion motivates the following construction of Verma module. We define formula_17 as the quotient vector space
where formula_113 is the left ideal generated by all elements of the form 
and
Because formula_113 is a left ideal, the natural left action of formula_84 on itself carries over to the quotient. Thus, formula_17 is a formula_84-module and therefore also a formula_3-module.
By extension of scalars.
The "extension of scalars" procedure is a method for changing a left module formula_121 over one algebra formula_122 (not necessarily commutative) into a left module over a larger algebra formula_123 that contains formula_122 as a subalgebra. We can think of formula_123 as a right formula_122-module, where formula_122 acts on formula_123 by multiplication on the right. Since formula_121 is a left formula_122-module and formula_123 is a right formula_122-module, we can form the tensor product of the two over the algebra formula_122:
Now, since formula_123 is a left formula_123-module over itself, the above tensor product carries a left module structure over the larger algebra formula_123, uniquely determined by the requirement that
for all formula_139 and formula_140 in formula_123. Thus, starting from the left formula_122-module formula_121, we have produced a left formula_123-module formula_134.
We now apply this construction in the setting of a semisimple Lie algebra. We let formula_146 be the subalgebra of formula_3 spanned by formula_5 and the root vectors formula_10 with formula_9. (Thus, formula_146 is a "Borel subalgebra" of formula_3.) We can form a left module formula_153 over the universal enveloping algebra formula_154 as follows:
The motivation for this formula is that it describes how formula_154 is supposed to act on the highest weight vector in a Verma module.
Now, it follows from the Poincaré–Birkhoff–Witt theorem that formula_154 is a subalgebra of formula_84. Thus, we may apply the extension of scalars technique to convert formula_153 from a left formula_154-module into a left formula_84-module formula_17 as follow:
Since formula_17 is a left formula_84-module, it is, in particular, a module (representation) for formula_3.
The structure of the Verma module.
Whichever construction of the Verma module is used, one has to prove that it is nontrivial, i.e., not the zero module. Actually, it is possible to use the Poincaré–Birkhoff–Witt theorem to show that the underlying vector space of formula_17 is isomorphic to
where formula_174 is the Lie subalgebra generated by the negative root spaces of formula_3 (that is, the formula_12's).
Basic properties.
Verma modules, considered as formula_3-modules, are highest weight modules, i.e. they are generated by a highest weight vector. This highest weight vector is formula_178 (the first formula_179 is the unit in formula_180 and the second is
the unit in the field formula_181, considered as the formula_146-module
formula_153) and it has weight formula_1.
Multiplicities.
Verma modules are weight modules, i.e. formula_17 is a direct sum of all its weight spaces. Each weight space in formula_17 is finite-dimensional and the dimension of the formula_37-weight space formula_188 is the number of ways of expressing formula_189 as a sum of positive roots (this is closely related to the so-called Kostant partition function). This assertion follows from the earlier claim that the Verma module is isomorphic as a vector space to formula_173, along with the Poincaré–Birkhoff–Witt theorem for formula_173.
Universal property.
Verma modules have a very important property: If formula_121 is any representation generated by a highest weight vector of weight formula_1, there is a surjective formula_3-homomorphism formula_195 That is, all representations with highest weight formula_1 that are generated by the highest weight vector (so called highest weight modules) are quotients of formula_197
Irreducible quotient module.
formula_17 contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight formula_199 If the highest weight formula_1 is dominant and integral, one then proves that this irreducible quotient is actually finite dimensional.
As an example, consider the case formula_201 discussed above. If the highest weight formula_63 is "dominant integral"—meaning simply that it is a non-negative integer—then formula_203 and the span of the elements formula_73 is invariant. The quotient representation is then irreducible with dimension formula_205. The quotient representation is spanned by linearly independent vectors formula_206. The action of formula_207 is the same as in the Verma module, "except" that formula_208 in the quotient, as compared to formula_209 in the Verma module.
The Verma module formula_17 itself is irreducible if and only if none of the coordinates of formula_1 in the basis of fundamental weights is from the set formula_212.
Other properties.
The Verma module formula_17 is called "regular", if its highest weight λ is on the affine Weyl orbit of a dominant weight formula_214. In other word, there exist an element w of the Weyl group W such that
where formula_216 is the affine action of the Weyl group.
The Verma module formula_17 is called "singular", if there is no dominant weight on the affine orbit of λ. In this case, there exists a weight formula_214 so that formula_219 is on the wall of the fundamental Weyl chamber (δ is the sum of all fundamental weights).
Homomorphisms of Verma modules.
For any two weights formula_220 a non-trivial homomorphism
may exist only if formula_37 and formula_1 are linked with an affine action of the Weyl group formula_224 of the Lie algebra formula_3. This follows easily from the Harish-Chandra theorem on infinitesimal central characters.
Each homomorphism of Verma modules is injective and the dimension
for any formula_227. So, there exists a nonzero formula_221 if and only if formula_188 is isomorphic to a (unique) submodule of formula_17.
The full classification of Verma module homomorphisms was done by Bernstein–Gelfand–Gelfand and Verma and can be summed up in the following statement:
 There exists a nonzero homomorphism formula_221 if and only if there exists 
a sequence of weights
such that formula_233 for some positive roots formula_234 (and formula_235 is the corresponding root reflection and formula_236 is the sum of all fundamental weights) and for each formula_237 is a natural number (formula_238 is the coroot associated to the root formula_234).
If the Verma modules formula_240 and formula_241 are regular, then there exists a unique dominant weight formula_214 and unique elements "w", "w"′ of the Weyl group "W" such that
and
where formula_216 is the affine action of the Weyl group. If the weights are further integral, then there exists a nonzero homomorphism
if and only if
in the Bruhat ordering of the Weyl group.
Jordan–Hölder series.
Let 
be a sequence of formula_3-modules so that the quotient B/A is irreducible with highest weight μ. Then there exists a nonzero homomorphism formula_246.
An easy consequence of this is, that for any highest weight modules formula_251 such that
there exists a nonzero homomorphism formula_246.
Bernstein–Gelfand–Gelfand resolution.
Let formula_254 be a finite-dimensional irreducible representation of the Lie algebra formula_3 with highest weight λ. We know from the section about homomorphisms of Verma modules that there exists a homomorphism
if and only if
in the Bruhat ordering of the Weyl group. The following theorem describes a resolution of formula_254 in terms of Verma modules (it was proved by Bernstein–Gelfand–Gelfand in 1975) :
There exists an exact sequence of formula_3-homomorphisms
where "n" is the length of the largest element of the Weyl group.
A similar resolution exists for generalized Verma modules as well. It is denoted shortly as the "BGG resolution".