Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group

http://dx.doi.org/10.4153/CJM-2005-024-7
Canad. J. Math. 57(2005), 598-615

  Published:2005-06-01
 Printed: Jun 2005

Differential operators $D_x$, $D_y$, and $D_z$ are formed using the action of the $3$-dimensional discrete Heisenberg group $G$ on a set $S$, and the operators will act on functions on $S$. The Laplacian operator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.