Gelfand-Tsetlin degeneration of shift of argument subalgebras in type D
The universal enveloping algebra of any semisimple Lie algebra contains a family of maximal commutative subalgebras, called shift of argument subalgebras, parametrized by regular Cartan elements of . For the Gelfand-Tsetlin commutative subalgebra in arises as some limit of subalgebras from this family. In our previous work [RZ18] we studied the analogous limit of shift of argument subalgebras for the Lie algebras and . We described the limit subalgebras in terms of Bethe subalgebras of twisted Yangians and , respectively, and parametrized the eigenbases of these limit subalgebras in the finite dimensional irreducible highest weight representations by Gelfand-Tsetlin patterns of types C and B. In this note we state and prove similar results for the last case of classical Lie algebras, . We describe the limit shift of argument subalgebra in terms of the Bethe subalgebra in the twisted Yangian and give a natural indexing of its eigenbasis in any finite dimensional irreducible highest weight -module by type D Gelfand-Tsetlin patterns.
To our teacher Rafail Kalmanovich Gordin
- 0 Introduction
- 1 Proofs of the main theorems.
- 2 Discussion
0.1 Commutative subalgebras in the universal enveloping algebra.
The universal enveloping algebra of any semisimple Lie algebra contains a family of maximal commutative subalgebras, called shift of argument subalgebras , parametrized by regular Cartan elements of . Their associated graded are subalgebras freely generated by the generators of centre () and all their derivatives along :
The latter are Poisson-commutative subalgebras of maximal possible transcendence degree, first introduced by Mishchenko and Fomenko in [MF79]. From the above description it follows that the Poincarè series of such algebras have the following form:
For the Gelfand-Tsetlin commutative subalgebra in arises as some limit of subalgebras from this family (see [Vin90, Shu02]). Its eigenbasis in every finite dimensional irreducible -module has a combinatorial description via the Gelfand-Tsetlin patterns (or GT-patterns of type A).
The analogous limits of shift of argument subalgebras exist for all classical Lie algebras. In [RZ18] we studied such limit subalgebras for and . Namely, we described the limit of the shift of argument subalgebra for in terms of Bethe subalgebras and in the twisted Yangians and , respectively (see Theorem A of loc. cit.). Here is the basis of the classical Lie algebra expressed in the standard matrix units as follows:
for , and
Moreover, we showed that the spectrum of such limit subalgebra in any irreducible finite dimensional -module is indexed by the datum called Gelfand-Tsetlin patterns of types B and C (see Theorem B of loc. cit). In this note we prove the analogous assertions for classical Lie algebras of the (last remaining) type D, i.e. for the case of .
0.2 Twisted Yangian .
Let us recall the definition of the twisted Yangian . For more detailed introduction to Yangians and their twisted versions we refer the reader to the classical paper by Molev, Nazarov, and Olshanski [MNO96].
The twisted Yangian is an algebra generated by the coefficients () of the power series
satisfying the following commutation relations:
The twisted Yangian is a coideal subalgebra in the usual Yangianof the Lie algebra .
There is a distinguished maximal commutative subalgebra called Bethe subalgebra. The subalgebra is generated by the elements () and the center of . For more general definition of Bethe subalgebras in twisted Yangians we refer the reader to [NO96].
The Olshanski centralizer construction gives an algebra homomorphism (see [RZ18, Theorem 2 (ii)], or [MO00, Propositions 4.14 and 4.15]). We denote by the commutative subalgebra of generated by and . The first result of the present note is the following description for the limit quantum shift of argument subalgebra , generalizing [RZ18, Theorem A] to classical Lie algebras of the type :
The limit of the quantum shift of argument subalgebra for in the universal enveloping algebra coincides with .
The idea of the proof is as follows. In [RZ18] we proved the inclusion for all classical types. Hence it remains to show that the Poincaré series of is greater or equal to that of for generic . Following the same strategy as in loc. cit. we get a lower bound for the Poincaré series of by presenting some explicit set of elements in which are algebraically independent and have the same degrees as the generators of . For the algebraic independence it is sufficient that the images of these elements in the associated graded algebra are algebraically independent. The latter is checked by taking the differentials of these elements at a regular nilpotent point and proving that these differentials are linearly independent. The two main differences from the analogous computation in the types B and C are, first, that here we have to consider Pfaffians of the successively embedded and compute their differentials, and, second, that contrary to the B and C cases, here the regular nilpotent element is not a maximal Jordan block, so the computation is different. This computation occupies the most part of the next section.
0.3 -module structure on multiplicty spaces and the eigenbasis of .
According to the Olshansky centralizer construction there is an irreducible highest weight -module structure on the multiplicity spaces of the restriction of any irreducible highest weight finite-dimensional -module to . Let us recall the description of this -module structure following [Mol06, Theorem 16 (ii)].
Let be the irreducible -module with the highest weight . Then we have a -module structure on
defined via the evaluation homomorphism and the comultiplication on . We also need the -dimensional -module () defined as
Let and be highest weights of finite-dimensional irreducible representations of and and denote the corresponding multiplicity space. Then the action of on defined as a composition of homomorphism to and a natural projection is irreducible and isomorphic to
where , ,
Consider the multiplicity space Note that the above module
has a (very naive) basis formed by the tensor products of the weight vectors in each tensor factor. This means that there is a basis of the multiplicity space indexed by collections of integers satisfying the following inequalities:
Iterating the above restriction procedure (similarly to the construction of the Gelfand-Tsetlin basis) we get a basis of irreducible -module corresponding to the combinatorial objects called D type pattern for
with and all entries are simultaneously from or , satisfying the following set of inequalities
By [HKRW17] the spectrum of limit algebra in any irreducible finite-dimensional -module is simple. Moreover, from Theorem 1 we know that the limit algebra is generated by commutative subalgebras in the successive centralizer algebras . Hence any eigenvector with respect to in has the following form: for some collection of highest weights of (with ) have where is an eigenvector of in the multiplicity space . Our purpose is to give a natural indexing of the (much more complicated) eigenbasis of with respect to by the same combinatorial data.
By Proposition 1 we can regard the eigenbasis for in as an element of the continuous family of eigenbases for in the tensor products with being free parameters and the differences being fixed integers. Then the following analog of [RZ18, Theorem B] holds in the type case (without any changes in the proof).
There is a path () in the space of parameters such that
is the collection of which occurs in our -module ;
the spectrum of on is simple for all ;
the limit of the corresponding eigenbasis as is just the product of the -weight bases in each .
We reproduce the construction of such path in the end of the next section. Once we have such we can take the parallel transport of the eigenbasis at to the product of the weight bases at thus obtain a bijection between the former and the latter. Note that the product of the weight bases of is naturally enumerated by the GT-patterns of the type D. So from the above Theorem we get the following
The spectrum of in any finite-dimensional irreducible -module of highest weight can be naturally parametrized by the set of Gelfand-Tsetlin type D patterns.
So the results of [RZ18] together with the present note can be summarized as follows:
Let be a Lie algebra of any classical type (, or ). The limit of the shift of argument subalgebra for in the universal enveloping algebra is . The spectrum of this subalgebra in any irreducible -module is simple and can be naturally identified with the set of Gelfand-Tsetlin patterns of the corresponding type.
The identification of the eigenbasis with the set of Gelfand-Tsetlin patterns depends on the choice of the path . In the last section of this note we present some arguments towards this identification being independent of this choice. The detailed proof based on these arguments is the subject of our future work. We also expect that this indexing of the eigenbasis agrees with the -crystal structure on the eigenbasis and on GT-patterns of classical types, defined in [HKRW17] and [Lit98], respectively (see [RZ18, Conjecture 2]).
The work of both authors has been funded by the Russian Academic Excellence Project ’5-100’. The first author has also been supported in part by the Simons Foundation.
1 Proofs of the main theorems.
1.1 An auxiliary lemma on the Pfaffian polynomial.
Recall that the determinant of a skew-symmetric matrix is the square of a polynomial called the Pfaffian of . This polynomial can be expressed in terms of matrix entries as follows:
where . We identify with by the unique order-preserving bijection and treat as the set of all bijective maps from to .
Introduce the following skew-symmetric element:
We have the following auxiliary result.
The differential of at is proportional to with a non-zero factor:
For we define a set of unordered pairs as:
Note that iff for all and either or . Hence we just need to examine the contribution to of all corresponding to satisfying the conditions above.
Assume and satisfies the conditions above. Then we have . To describe such ’s we need to fix the order in each pair of indices (note that the choice of the order does not change the value of since changing the order changes both the parity of the permutation and the sign of ). There is also an action of on (simply permuting the pairs) and this action preserves the property of satisfying the required condition. We observe that does not change if we act with a transposition on . Indeed, both the parity of the permutation and the monomial do not change. This shows that all the ’s with are equal.
Multiplying by the transposition of 1 and -1 is a bijection between the sets of ’s with and with . This means that each with has a contribution opposite to the one for ’s with . This implies the assertion of the Lemma. ∎
1.2 Algebraic independence of some elements of
Introduce the principal nilpotent element of :
Let , , , where
and ; .
Consider the following total order on the set :
We extend this order to a partial order on the basis of by simply setting each not from to be less than any element of . Now we are ready to prove the following lemma.
The differentials of the elements , () and at have the following form (here denotes proportionality):
if is even:
if is odd:
Among all differentials , and there is only which has a nonzero coefficient at . The only monomials with a nonzero contribution to at are
Since their degrees are equal to they can appear only in . A similar situation occurs for and the corresponding monomials are:
After writing down the differential at all of the above monomials have the same sign before (we make use of ) hence the contribution is nonzero.
Analogously we prove all the statements till the point when they differ for odd and even values of . In this proof we just consider the case of an even (the odd case can be proved identically). If we look at then the only monomials which can contribute to it after differentiating at are:
For the latter monomial we again notice . Since is even their differentials have the same sign at . Both of them require hence it must be . But at the same time has a nonzero coefficient before because of the following monomials:
It is easy to see that both and appear in with the same nonzero coefficient.
Now we consider the Pfaffian-type element . The -matrix becomes skew-symmetric under the transformation
where is the skew-symmetric image of -matrix (the sign depends on the parity of ). From the Lemma 1 it immediately follows that .
The remaining statements can be verified in a similar manner. ∎
This lemma implies the next statement.
The elements , and for and are algebraically independent elements of .
From the description of the differentials of these elements at the principal nilpotent element provided by Lemma 2 for we see that they are linearly independent, hence the Lemma follows. ∎
1.3 The proof of Theorem 1.
In Proposition 5 ([RZ18]) we showed that . So it is sufficient to show that the subalgebras and have the same Poincaré series.
The Poincaré series for coincides with such for with generic , hence, equals to:
On the other hand, the subalgebra contains the elements (of the degree ), as well as the central generators (of the degree ), for all , , and the symmetrization of the Pfaffians (of the degree ) for all (here is the symmetrization map from the symmetric algebra associated with to the universal enveloping algebra). So for proving the equality of the it remains to check that the above elements of are algebraically independent. Moreover, it is sufficient to show that the images of these elements in the associated graded algebra are algebraically independent. Note that the latter follows from Lemma 3: indeed, we have , and .
1.4 The indexing of the spectrum by GT-patterns.
First, we note that the proof of Theorem B in [RZ18] works for Theorem 2 of the present paper without any changes.
Recall that -module is defined for , as:
where ’s are free parameters and is the 1-dimensional representation of introduced before. Take where are fixed real numbers and . Let be the image of in . The images of generators of in the space of endomorphisms depend on ’s. We can replace the generators () of with . The algebra generated by them still coincide with but now we can consider the desired limit by making . Moreover, in [RZ18, Corollary 1] we showed that the limit of the eigenbasis as ( ()) coincides with the product of weight bases for each ().
From the definition of it follows that for every we have
For generic subalgebra acts on with simple spectrum. On the other hand, the limit shift of argument subalgebra acts with simple spectrum on the multiplicity space (see [HKRW17]). The latter is identified with when all .
Hence, by choosing an appropriate path connecting with the limit at , the basis of can be naturally indexed by the collections of integers () such that . So it means that we have
Continuing this restriction further we get the parametrization of the spectrum with Gelfand-Tsetlin patterns of type D.
In [RZ18, Conjecture 1] we conjectured that the indexing of the eigenvectors of in does not actually depend on the choice of the path connecting to . Here we present some speculation on this conjecture.
The property that has simple spectrum in is in fact the intersection of the following two Zariski open conditions on :
has a cyclic vector in ;
acts on semisimply.
For our conjecture, it is sufficient to show that the complement of this subset is of complex codimension in some simply connected neighborhood of in . Indeed, this will mean that the spectrum of in is an unramified covering of a simply connected real manifold outside real codimension , hence this covering is in fact trivial. So the parallel transport of the eigenvectors along any path connecting given points in (and avoiding the “bad” points) is the same.
First, one can show that for the algebra acts on