Special function models of indecomposable $sl(2)$ representations: The Laguerre Case

Lie algebras, their realizations and connections with special functions have attracted a lot of attention in the area of mathematical physics since the end of the 60s [1, 2, 3, 4, 5, 6, 7]. This has been inspired partly by the fruitful connection of special functions and Lie algebras with quantum mechanical problem [8, 9, 10]. In this context, the generators of the Lie algebras can be integrals of motion, ladder operators or other physically relevant operators. Via the notion of hidden symmetry underlying exact and quasi-exact solvability, the $sl(2)$ Lie algebra was connected to various models [11]. The realizations related to other higher-rank Lie algebras such as $sl(n)$ and $so(n)$ have lead to profound connection with generalized hypergeometric, Lauricella and Horn functions (see e.g. [2]). Those special functions have appeared in different contexts of mathematical physics.

The cases of the Lie algebras $sl(2)$ with one and two variables realizations have been intensively described in regard of unitary and irreducible representations of $sl(2)$. This rank-1 algebras has been related to different orthogonal polynomials, in particular using first but also second-order differential operators [12, 13, 14, 15, 16, 17, 18]. An extensive description of Lie algebras and their realizations, and in particular from point of view of the Casimir invariant was developed by Vilenkin and Klimyk [19]. Some connections with wider types of representation beyond irreducible and unitary was pointed out [20], in regard of indecomposable representations [21, 22, 23] and quantum field theory [24].

Among the indecomposable representations, there are different cases that consist in combinations of finite and infinite dimensional representations glued together via the action of the Cartan or other generators and forming interconnected chains. The paper intends to generalize certain indecomposable representations in context of $sl(2)$ with arbitrary number of chains from the perspective of differential operators that were obtained in context of the Laguerre case[2]. This will lead to constructing various polynomials where the raising generators of $sl(2)$ will act almost as usual ladder operators but allowing interaction between the different chains. The papers will also establish the connection with systems of coupled nonhomogeneous second-order differential equations.

Standard literature on the theory of ODEs and special functions has mostly looked at uncoupled homogeneous ODEs in the hypergeometric class and their generalisation as the Heun equation [25, 26]. The study of systems of non-homogeneous differential equations, their solutions and connections to special functions have been much less studied. Some cases related to resonant problem [28, 29, 30, 31, 32] were studied. However, there is no systematic search for solution of those systems of ODEs. This paper allows to discuss some classes of such systems of ODEs, that naturally extend the ODE of the Laguerre polynomials, and with an underlying Lie algebras such as sl(2) and related indecomposable representations. This makes those polynomials and special functions of interest for applications in mathematical physics. Here the connection between indecomposable representations and $sl(2)$ provides a new perspective on this topic.

The papers is organised as follow. In Section 2, we set the foundation and notation of the manuscript while reviewing properties of the (generalized) Laguerre polynomials. In Section 3 we present the case of indecomposable with chains where T, the Cartan generator, act as a Jordan block. We also present explicitly the polynomials solving the systems of nonhomogenous coupled ODEs. We give extra details on the 2 and 3 chains problems. In particular we present second-order differential-recurrence relations, generating functions and formula in terms of linear combination of Laguerre polynomials. In Section 4 we characterise the case of Laguerre polynomials where T is diagonalizable (but the raising/lowering operators are not). In Section 5 we provide further insight into some of the building block taking the form of new polynomials. We present the third and fourth order ODE that they satisfy, further recurrence relations, related family of polynomial and bi-orthogonality type relations and insight into the zeroes of those polynomials. In Section 6, we discuss how those are in fact connected with certain representations of $sl(2)$ where the infinite dimensional representation is obtained via induced representation of a finite representation of the Borel subalgebra.

The motivation for the treatment of indecomposable representations presented here takes origin in the large body of work on $sl(2)$ and what we would refer as one-chain representations, and studied in the monograph [2] in which the differential recurrence relations for hypergeometric, confluent hypergeometric and Bessel functions were shown to be expressible as representations of the three four-dimensional Lie algebras $sl(2)\oplus(E)$, the Heisenberg algebra, and $e(2)\oplus(E)$. The differential recurrence relations for these three types of hypergeometric functions are linear first-order ordinary differential equations for the functions $F_{n}(z)$. The idea is to add a new variable $t$ and to replace the ordinary differential equations by partial differential equations in $z,t$ such that in the new system the basis functions are $F_{n}(z)t^{n}$. If this is done correctly the new recurrence operators will close to generate one of the three Lie algebras listed above. The advantage of the addition of a 2nd variable is that the tools of Lie theory become available and we can find new properties of these functions, such as generating functions with relative ease.

In the monograph [2] these realizations were used to classify the finite and infinite dimensional bounded above or below irreducible representatives of $sl(2)\oplus(E)$, the Heisenberg algebra, and $e(2)\oplus(E)$, most of which do not extend to global representations of the corresponding groups. However, local Lie group theory still applies and can be used to derive additional theorems and generating functions where global group representations no longer apply.

In this paper we shall employ the realization of type B ($sl(2)$, Laguerre polynomials) of the differential recurrence relations employed in [2], but now used to find realizations of indecomposable representations of $sl(2)$.

If we restrict ourselves to one-chain in the type B, the (generalized) Laguerre are known to possess raising and lowering operators forming an $sl(2)$ algebra, that is

where $E^{+}$ and $E^{-}$ are respectively the raising and lowering operators. The $\mathfrak{sl}_{2}$ algebra possess the following Casimir element:

In this case, we will consider that none of the operators act centrally. One representation of such an algebra takes the form

i.e. $T$ acts centrally and north, $E^{+}$ acts east and $E^{-}$ acts west and north-west. It is important to note that on the highest chain, all the operators act like the operators in equation (2.4). Let us denote the vanishing of the north coefficients on the highest chain. A realization is of this representation is given by

which also satisfy the $sl(2)$ commutation relations (2.1).

To generate a new chain, one can use the equation

Using the realization of $T$ and considering $|0,N-1\rangle$ as an unknown function of $z$ and $t$, we obtain a linear differential equation that we can solve. We can then generate all the other elements of the chain by successively applying $E^{+}$ on each state $|n,N-1\rangle$. Finally, by checking the restriction of $sl(2)$, we can construct the whole states on the second chain.

Once the second chain is constructed, we can reapply the same method to construct the third (or lower) chains. As a result, we can express all the states as

where $n$ is any natural number and $\beta\in\{1,...,N\}$. The functions $\omega_{n,j}(z)$ are polynomials in $z$ satisfying the raising and lowering relations

By combining those two last equations, we can construct a linear second-order differential equation

which is the generalized Laguerre differential equation (2.7) but with a non-homogeneous term. For the highest chain, we have by construction that $\omega_{n,N}(z)$ is the generalized Laguerre polynomials, i.e.

Note that any polynomial $\omega_{n,\beta}(z)$ where $n<0$ or $\beta<N$ is set to zero.

Using the representation above, the Casimir element $\mathcal{K}$ in equation (2.2) does not act centrally, that is

By applying $E^{+}$ repetitively on a state, we have

Therefore, if we exponentiate the raising operator $E^{+}$ and apply it on a state, we obtain

If we take $n=0$, $\beta=N$ and $u=1$, then we obtain the equivalent of the generating function of the Laguerre polynomial

In fact, applying the exponentiation of $E^{+}$ on any locally analytic function $f(z,t)$ (see [2]) , we have that

In the case where $T$ can be diagonalized, we can consider the actions of $T$, $E^{+}$ and $E^{-}$ as

i.e. where $T$ acts centrally, $E^{+}$ acts east and $E^{-}$ acts west and north-west. A realization of this action is given the same realization as equation (3.5), that is

Conversely to the previous case, we will consider that the lowest weight going west is not when $n=0$ but when $n=N-\beta$ for a state $|n,\beta\rangle$ for a system with $N$ state. For a finite number of chains, this arrangement of states takes the form of an infinite trapezoid.

For this representation, we cannot use $T$ to solve for the first state of the second chain, i.e. $|1,N-1\rangle$, but we can use can use $E^{-}$. We have

N.B. that the first chain is known and given by $|n,N\rangle=L_{n}^{\alpha}(z)t^{n}$. Once we the state $|1,N-1\rangle$, we can generate all the other states using $E^{+}$. By checking all the commutation relations of $sl(2)$, and then generating $N$ chains, we obtain

where $\omega_{n,\beta}(z)$ is polynomial in $z$. Those polynomials $\omega_{n,\beta}(z)$ satisfy the raising and lowering recurrence relations

By combining those two last equations, we can construct a linear second-order differential equation

The Casimir acts north and centrally, that is

By using $E^{+}$, it is possible to construct a generating function. We have

Hence,

By taking $u=1$, $n=0$ and $\beta=N$ and using previous results, we get

which matches Laguerre’s generating function.

In the construction of those indecomposable representations, there are various special functions, first at the level of the two variables differential operators satisfying an $sl(2)$ Lie algebra which involve the element of $V_{n}$ that take the form of two-variables special functions. The $\omega_{n,i}$ are building block of the representations for the diagonal and nondiagonal cases. Those $\omega_{n,i}$ also satisfy various differential-recurrence relations based on the raising and lowering operators $E^{+}$ and $E^{-}$. We have also provided generating functions and explicit solutions in terms of linear combinations of Laguerre orthogonal polynomials which allow to get further insight into their properties. This section will be devoted to provide ordinary differential equations, further recurrence relations and properties such as bi-orthogonality type relations with another set of related polynomial and zeroes of those polynomials in the case of the two chains.