# Orthogonal Stochastic Duality Functions from Lie Algebra Representations

## Abstract

We obtain stochastic duality functions for specific Markov processes using representation theory of Lie algebras. The duality functions come from the kernel of a unitary intertwiner between \(*\)-representations, which provides (generalized) orthogonality relations for the duality functions. In particular, we consider representations of the Heisenberg algebra and \(\mathfrak {su}(1,1)\). Both cases lead to orthogonal (self-)duality functions in terms of hypergeometric functions for specific interacting particle processes and interacting diffusion processes.

## Keywords

Stochastic duality Lie algebra representations Hypergeometric functions Orthogonal polynomials## 1 Introduction

A very useful tool in the study of stochastic Markov processes is duality, where information about a specific process can be obtained from another, dual, process. The concept of duality was introduced in the context of interacting particle systems in [17], and was later on developed in [15]. For more applications of duality see e.g. [4, 6, 13, 18].

Two processes are in duality if there exists a duality function, i.e. a function of both processes such that the expectations with respect to the original process is related to the expectations with respect to the dual process (see Sect. 2 for a precise statement). Recently in [5, 16] orthogonal polynomials of hypergeometric type were obtained as duality functions for several families of stochastic processes, where the orthogonality is with respect to the corresponding stationary measures. These orthogonal polynomials contain the well-known simpler duality functions (in the terminology of [16], the classical and cheap duality functions) as limit cases. In [5], Franceschini and Giardinà use explicit relations between orthogonal polynomials of different degrees, such as raising and lowering formulas, to prove the stochastic duality. In [16], Redig and Sau find the orthogonal polynomials using generating functions. With a similar method they also obtain Bessel functions, which are not polynomials, as self-duality function for a continuous process.

The goal of this paper is to demonstrate an alternative method to obtain the orthogonal polynomials (and other ‘orthogonal’ functions) from [5, 16] as duality functions. The method we use is based on representation theory of Lie algebras. This is inspired by [3, 7], where representation theory of \(\mathfrak {sl}(2,{\mathbb {C}})\) and the Heisenberg algebra is used to find (non-orthogonal) duality functions, see also Sturm et al. [19] for a Lie algebraic approach to duality. Roughly speaking, the main idea is to consider a specific element *Y* in the Lie algebra (or better, enveloping algebra). Realized in two different, but equivalent, representations \(\rho \) and \(\sigma \), \(\rho (Y)\) and \(\sigma (Y)\) are the generators of two stochastic processes. In case of \(\mathfrak {sl}(2,{\mathbb {C}})\), *Y* is closely related to the Casimir operator. The duality functions come from an intertwiner between the two representations. In this paper we consider a similar construction with unitary intertwiners between \(*\)-representations, so that the duality functions will satisfy (generalized) orthogonality relations.

In Sect. 2 the general method to find duality functions from unitary intertwiners is described. In Sect. 3 the Heisenberg algebra is used to show duality and self-duality for the independent random walker process and a Markovian diffusion process. The self-duality of the diffusion process seems to be new. The (self-)duality functions are Charlier polynomials, Hermite polynomials and exponential functions. In Sect. 4 we consider discrete series representation of \(\mathfrak {su}(1,1)\), and obtain Meixner polynomials, Laguerre polynomials and Bessel functions as (self)-duality functions for the symmetric inclusion process and the Brownian energy process. We would like to point out that the self-duality functions are essentially the (generalized) matrix elements for a change of base between bases on which elliptic or parabolic Lie group / algebra elements act diagonally, see e.g. [2, 14], so in these cases stochastic self-duality is a consequence of a change of bases in the representation space.

### 1.1 Notations and Conventions

By \({\mathbb {N}}\) we denote the set of nonnegative integers. We often write *f*(*x*) for a function \(x\mapsto f(x)\); the distinction between the function and its values should be clear from the context. For functions \(x\mapsto f(x;p)\) depending on one or more parameters *p*, we often omit the parameters in the notation. For a set *E*, we denote by *F*(*E*) the vector space of complex-valued functions on *E*. \({\mathcal {P}}\) is the vector space consisting of polynomials in one variable.

*i*, then the series is a finite sum.

## 2 Stochastic Duality Functions from Lie Algebra Representations

In this section we describe the method to obtain stochastic duality functions from \(*\)-representations of a Lie algebra. This method will be applied in explicit examples in Sects. 3 and 4 .

### 2.1 Stochastic Duality

### 2.2 Lie Algebra Representations

*F*(

*E*). We call \(\rho \) a \(*\)-representation of \({\mathfrak {g}}\) on \({\mathcal {H}}=L^2(E,\mu )\) with dense domain \({\mathcal {D}} \subseteq {\mathcal {H}}\), if \(\rho (X)\) is defined on \({\mathcal {D}}\) and \(\langle \rho (X)f,g\rangle = \langle f, \rho (X^*) g \rangle \) for all \(X \in {\mathfrak {g}}\) and all \(f,g \in {\mathcal {D}}\). A \(*\)-representation of \({\mathfrak {g}}\) extends uniquely to a \(*\)-representation of \(U({\mathfrak {g}})\) on \({\mathcal {H}}\) (with the same domain \({\mathcal {D}}\)).

Two \(*\)-representations \(\rho _1\) and \(\rho _2\) are unitarily equivalent if there exists a unitary operator \(\Lambda :{\mathcal {H}}_1 \rightarrow {\mathcal {H}}_2\) such that \(\Lambda ({\mathcal {D}}_1) = {\mathcal {D}}_2\) and \(\Lambda [\rho _1(X) f] = \rho _2(X) \Lambda (f)\) for all \(X \in {\mathfrak {g}}\) and \(f \in {\mathcal {D}}_1\).

### Lemma 2.1

- 1.
\([\rho _1(X^*)K(\cdot ,y)](x) = [\rho _2(X) K(x,\cdot )](y)\) for all \(X \in {\mathfrak {g}}\) and \((x,y) \in E_1 \times E_2\).

- 2.The operator \(\Lambda :{\mathcal {D}}_1 \rightarrow L^2(E_2,\mu _2)\) defined byextends to a unitary operator \(\Lambda : L^2(E_1,\mu _1) \rightarrow L^2(E_2,\mu _2)\).$$\begin{aligned} \Lambda f = \left( y \mapsto \int _{E_1} f(x) K(x,y)\, d\mu _1(x) \right) , \end{aligned}$$

### Proof

### 2.3 Duality from \(*\)-Representations

We are now ready to obtain duality functions for certain operators from the kernels of intertwining operators between \(*\)-representations.

### Theorem 2.2

### Proof

### Remark 2.3

*Y*, which is self-adjoint in \(U({\mathfrak {g}})^{\otimes N}\), will always have a specific form corresponding to (2.1);

*i*th factor and \(X_{(2)}\) in the

*j*th factor. In fact, we obtain duality between the operators \(\rho ({\hat{Y}}_{i,j})\) and \(\sigma ({\hat{Y}}_{i,j})\) corresponding to each of the terms of the sum in (2.3).

## 3 The Heisenberg Algebra

To illustrate how the method from the previous section is applied, we use the Heisenberg Lie algebra to obtain duality functions for two stochastic processes. Let us first describe the processes.

*N*sites, and each site can contain an arbitrary number of particles. Particles jump from site

*i*to site

*j*with rate proportional to the number of particles \(n_i\) at site

*i*. Let \(p_{i,j}\ge 0\). The generator of this process is the difference operator acting on appropriate function in \(F({\mathbb {N}}^N)\) given by

*i*th component and all other component are 0.

*N*Brownian motions which are attracted to each other with a rate proportional to their distances. The generator is a differential operator on appropriate functions in \(F({\mathbb {R}}^N)\) given by

Note that both generators have the form (2.1).

### Lemma 3.1

### Proof

### Remark 3.2

We can also consider the tensor product representation \(\rho _{c_1} \otimes \cdots \otimes \rho _{c_N}\) with (possibly) \(c_i \ne c_j\). This leads to a generator of a Markov process depending on *N* different parameters. However, to prove self-duality it seems crucial to assume \(c_i=c\) for all *i*, see Lemma 3.5 later on.

### 3.1 Charlier Polynomials and Self-Duality of IRW

*a*and \(a^\dagger \) on the Charlier polynomials

*C*(

*n*,

*x*;

*c*) as a kernel later on. For notational convenience we will often omit the dependence on

*c*in the notation; \(C(n,x)=C(n,x;c)\). Using the raising and lowering properties (3.6) we obtain

*x*-variable are similar to the actions of \(Z-a\) and \(Z-a^\dagger \) in the

*n*-variable. This motivates the definition of the following isomorphism.

### Lemma 3.3

### Proof

The proof consists of checking the commutation relations (3.3), which is a straightforward computation. \(\square \)

### Proposition 3.4

*C*(

*n*,

*x*) satisfies

### Proof

*C*(

*m*,

*x*) is \(\Vert C(m,\cdot )\Vert ^2 = \frac{1}{w_c(m)}\). So \(\Lambda \) maps an orthogonal basis to another orthogonal basis with the same norm, hence \(\Lambda \) is unitary.

*Z*is clear. Now the result follows from the definition of \(\theta \), see Lemma 3.3.

\(\square \)

We are almost ready to prove self-duality for IRW, but first we need to know the image of *Y*, see (3.5), under the isomorphism \(\theta \otimes \theta \).

### Lemma 3.5

*R*is the zero operator on \({\mathcal {H}}_c \otimes {\mathcal {H}}_c\).

### Proof

After a somewhat tedious computation using the definition of \(\theta \) in Lemma 3.3, we find the explicit expression for \(\theta \otimes \theta (Y)\). Using \(\rho _c(Z) = c\, \mathrm {Id}\) it follows that \(\rho _c \otimes \rho _c (R) = 0\). \(\square \)

We can now apply Theorem 2.2 with \(\rho =\rho _{c} \otimes \cdots \otimes \rho _{c}\) and \(\sigma =\rho \circ (\theta \otimes \cdots \otimes \theta )\). Using Lemma 3.5 we find \(\sigma (Y_{i,j})=\rho (Y_{i,j})\), and then it follows that \(\sum p_{i,j} \sigma (Y_{i,j}) = L^{\mathrm {IRW}}\), see Lemma 3.1. So we obtain the well-known self-duality of the independent random walker process. Here the duality function is a product of Charlier polynomials.

### Theorem 3.6

### Remark 3.7

The intertwining operator \(\Lambda : {\mathcal {H}}_c \rightarrow {\mathcal {H}}_c\) maps the orthogonal basis of cheap duality functions to an orthogonal basis of Charlier polynomials (see the proof of Proposition 3.4), so \(\Lambda \) can be considered as a change of basis for \({\mathcal {H}}_c\). In this sense the self-duality functions in Theorem 3.6 are the matrix elements of a change of base in \({\mathcal {H}}_c^{\otimes N}\).

### 3.2 Hermite Polynomials and Duality Between IRW and the Diffusion Process

*a*and \(a^\dagger \) as differential operators. We define

### Lemma 3.8

*H*(

*n*,

*x*) form an orthogonal basis for \(\mathrm H_c\), with squared norm \(\Vert H(n,\cdot )\Vert ^2 = \frac{1}{w_c(n)}\). We define the representation \(\sigma _c\) by

### Proposition 3.9

*H*(

*n*,

*x*) satisfies

### Proof

Unitarity of \(\Lambda \) is proved in the same way as in Proposition 3.4. The intertwining property for the kernel follows from Lemma 3.8. Lemma 2.1 then shows that \(\Lambda \) intertwines \(\rho _c\) and \(\sigma _c\). \(\square \)

Similar as in Lemma 3.1 we find that the generator \(L^{\mathrm {DIF}}\) defined by (3.2) is the realization of *Y* defined by (3.5) on the Hilbert space \(\mathrm H_{c}^{\otimes N}\).

### Lemma 3.10

Finally, applying Theorem 2.2 we obtain duality between \(L^{\mathrm {IRW}}\) and \(L^{\mathrm {DIF}}\), with duality function given by Hermite polynomials.

### Theorem 3.11

### Remark 3.12

This duality between \(L^{\mathrm {IRW}}\) and \(L^{\mathrm {DIF}}\) was also obtained in [7, Remark 3.1], but Hermite polynomials are not mentioned there. Hermite polynomials of even degree have appeared as duality functions in [5, §4.1.1]; this can be considered as a special case of duality involving Laguerre polynomials, see [5, §4.2.1] or Theorem 4.15.

### 3.3 The Exponential Function and Self-Duality of the Diffusion Process

To show self-duality of the differential operator \(L^{\mathrm {DIF}}\), the following isomorphism is useful.

### Lemma 3.13

### Proof

We just need to check commutation relations, which is a direct calculation. \(\square \)

### Lemma 3.14

### Proof

*x*and

*y*, we find

Now we can show that the integral operator with \(\phi \) as a kernel is the desired intertwining operator.

### Proposition 3.15

### Proof

By applying Theorem 2.2 and using Lemma 3.10 we obtain self-duality of \(L^{\mathrm {DIF}}\) (3.2).

### Theorem 3.16

## 4 The Lie Algebra \(\varvec{\mathfrak {su}}(1,1)\)

*k*), \(k \in {\mathbb {R}}_{>0}^N\), which is a Markov jump process on

*N*sites, where each site can contain an arbitrary number of particles. Jumps between two sites, say

*i*and

*j*, occur at a rate proportional to the number of particles \(n_i\) and \(n_j\). Let \(p_{i,j}\ge 0\). The generator of this process is given by

*k*), \(k \in {\mathbb {R}}_{>0}^N\), which is a Markov diffusion process that describes the evolution of a system of

*N*particles that exchange energies. The energy of particle

*i*is \(x_i>0\). The generator is given by

*H*,

*E*,

*F*with commutation relations

### Remark 4.1

*c*is a fixed parameter, and just write \(\pi _k\) and \({\mathcal {H}}_k\) instead of \(\pi _{k,c}\) and \({\mathcal {H}}_{k,c}\).

### Lemma 4.2

### Proof

*a*this is either an elliptic element (\(|a|>1\)), parabolic element (\(|a|=1\)), or hyperbolic element (\(|a|<1\)), corresponding to the associated one-parameter subgroups in \(\mathrm {SU}(1,1)\).

### 4.1 Meixner Polynomials and Self-Duality for SIP

*x*-variable for the Meixner polynomials.

### Lemma 4.3

### Proof

This follows from the three-term recurrence relation for the Meixner polynomials.

\(\square \)

Using the difference equation for the Meixner polynomials, we can realize *H* as a difference operator acting on *M*(*n*, *x*) in the *x*-variable.

### Lemma 4.4

### Proof

*M*(

*n*,

*x*), which is, by self-duality, equivalent to the three-term recurrence relation:

With the actions of \(X_{a(c)}\) and *H* on Meixner polynomials, it is possible to express *E* and *F* acting on *M*(*n*, *x*) as three-term difference operators in the variable *x*. This leads to a representation by difference operators in *x*, in which the basis elements *H*, *E* and *F* all act by three-term difference operators. Having actions of *H*, *E* and *F*, we can express, after a large computation, \(\Delta (\Omega )\) in terms of difference operators in two variables \(x_1\) and \(x_2\). We prefer, however, to work with a simpler representation in which *H* acts as a multiplication operator, and *E* and *F* as one-term difference operators. Note that the action of \(X_{a(c)}\) in the *x*-variable corresponds up to a constant to the action of *H* in the *n*-variable, i.e. it is a multiplication operator. We can make a new \(\mathfrak {sl}(2,{\mathbb {C}})\)-triple with \(X_{a(c)}\) playing the role of *H*. The following result from [8, §3.2], where it is proved using conjugation with a group element, gives the corresponding isomorphism.

### Lemma 4.5

### Proof

Note that \(\theta _c\) preserves the \(*\)-structure, i.e. \(\theta _c(X^*) = \theta _c(X)^*\).

*H*. This allows us to write down explicit actions of \(E_c\) and \(F_c\) acting on

*M*(

*n*,

*x*) in the

*x*-variable. This then shows that

*M*(

*n*,

*x*) has the desired intertwining properties.

### Lemma 4.6

*M*(

*n*,

*x*) satisfy

### Proof

*H*we find

Now we are ready to define the intertwiner.

### Proposition 4.7

*M*(

*n*,

*x*) satisfies

### Proof

Unitarity follows from the orthogonality relations and completeness of the Meixner polynomials. The properties of the kernel follow from Lemma 4.6. \(\square \)

*k*).

### Theorem 4.8

### Remark 4.9

The Lie algebra \(\mathfrak {su}(2)\) is \(\mathfrak {sl}(2,{\mathbb {C}})\) equipped with the \(*\)-structure defined by \(H^*=H, E^*=F\). It is well known that \(\mathfrak {su}(2)\) has only finite dimensional irreducible \(*\)-representations. These can formally be obtained from the \(\mathfrak {su}(1,1)\) discrete series representation (4.4) by setting \(k=-j/2\) for some \(j \in {\mathbb {N}}\), where \(j+1\) is the dimension of the corresponding representation space. If we make the corresponding substitution \(k_i=-j_i/2\) in the generator (4.1) of the symmetric inclusion process, we obtain the generator of the symmetric exclusion process SEP on *N* sites where site *i* can have at most \(j_i\) particles. Making a similar substitution in Theorem 4.8 we find self-duality of SEP, with duality function given by a product of Krawtchouk polynomials.

### 4.2 Laguerre Polynomials and Duality Between SIP and BEP

### Lemma 4.10

Just as we did in the elliptic case, we can define an algebra isomorphism that will be useful. In this case, the element \(X_1\) corresponds to the generator *E*.

### Lemma 4.11

### Lemma 4.12

*L*(

*n*,

*x*) satisfy

### Proof

*H*,

*E*,

*F*by

### Proposition 4.13

*L*(

*n*,

*x*) satisfies

### Lemma 4.14

### Proof

Finally, application of Theorem 2.2 gives duality between the symmetric inclusion process SIP(*k*) and the Brownian energy process BEP(*k*).

### 4.3 Bessel Functions and Self-Duality of BEP

*f*, and the inverse is given by \({\mathcal {F}}_\nu ^{-1} = {\mathcal {F}}_\nu \).

*H*and

*E*on the eigenfunctions.

### Lemma 4.16

### Proof

*F*follows from the differential equation for the Bessel functions. We have

*x*and

*y*, we obtain the action of

*E*. Finally, having the actions of

*E*and

*F*, we find the action of

*H*from \(H=[E,F]\). \(\square \)

Using the Hankel transform we can now define a unitary intertwiner with a kernel that has the desired properties.

### Proposition 4.17

### Proof

Since the set of polynomials \({\mathcal {P}}\) is dense in \(\mathrm H_k\), it is enough to define \(\Lambda \) on \({\mathcal {P}}\). Unitarity of \(\Lambda \) is essentially unitarity of the Hankel transform \({\mathcal {F}}_{2k-1}\). The intertwining property follows directly from Lemma 4.16. \(\square \)

Note that \(\Lambda \) intertwines between \(*\)-representation with respect to the \(*\)-structure given by (4.9). Equivalently, \(\Lambda \) intertwines \(\sigma \circ \theta ^{-1}\) with itself, which is a \(*\)-representation with respect to the \(\mathfrak {su}(1,1)\)-\(*\)-structure.

Let \(k \in {\mathbb {R}}_{>0}^N\), and consider the tensor product representation \(\sigma _k\) defined by (4.11). Then from Proposition 4.17, Lemma 4.14 and Theorem 2.2 we obtain self-duality of the Brownian energy process BEP(*k*).

### Theorem 4.18

### 4.4 More Duality Relations

The Meixner polynomials from Sect. 4.1 can be considered as overlap coefficients between eigenvectors of the elliptic Lie algebra element *H* and another elliptic element \(X_{a(c)}\). There is a similar interpretation as overlap coefficients for the Laguerre polynomials (elliptic *H* - parabolic \(X_1\)) and Bessel functions (parabolic \(X_1\) - parabolic \(X_{-1}\)). So far we did not consider overlap coefficients involving a hyperbolic Lie algebra element, because there does not seem to be an interpretation in this setting for the element *Y* from (4.5) as generator for a Markov process. However, the construction we used still works and leads to duality as operators between \(L^{\mathrm {SIP}}\) or \(L^{\mathrm {BEP}}\) and a difference operator \(L^{\mathrm {hyp}}\) defined below, which may be of interest. We will give the main ingredients for duality between \(L^{\mathrm {SIP}}\) and \(L^{\mathrm {hyp}}\) in case \(N=2\) using overlap coefficients between elliptic and hyperbolic bases, which can be given in terms of Meixner–Pollaczek polynomials, see also [9, 12].

*Y*is given by (4.5), then we see that \(L^{\mathrm {hyp}}\) is the difference operator given by

*P*(

*x*,

*n*) as a kernel, but we do not actually need the intertwiner, since the kernel is enough to state the duality result. Using \(\theta _\phi (\Omega )=\Omega \) we obtain duality between the operators \(L^{\mathrm {SIP}}\) and \(L^{\mathrm {hyp}}\), with duality function given by the product

## Notes

### Acknowledgements

I thank Gioia Carinci, Chiara Franceschini, Cristian Giardinà and Frank Redig for very helpful discussions and giving valuable comments and suggestions.

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