This appendix is the manual for a set of functions written to assist the reader to understand and apply the theorems on GL(n, R) set out in the main part of the book. The software for the package is provided over the world wide web at http://www.math.waikato.ac.nz/∼kab and is in the form of a standard Mathematica add-on package. To use the functions in the package you will need to have a version of Mathematica at level 4.0 or higher. §A.2 Functions for GL(n)pack ApplyCasimirOperator[m,expr,iwa]

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... (aco): The operator acts on an expression. BruhatCVector[a] (bcv): The minors (c 1 , .., c n−1 ). BruhatForm[a] (bru): The four Bruhat factors of a non-singular symbolic matrix. BlockMatrix[a,rows,cols] (blm): Extract a general sub-block of a matrix. CartanForm[a] (car): The two Cartan factors of a numeric nonsingular matrix. ConstantMatrix[c,m,n] (com): Construct a constant matrix of given size. CreQ[e] (crq): Check a Canonical Rational Expression. DiagonalToMatrix[d] (d2m): Convert a list to a diagonal matrix. EisensteinFourierCoefficient[z,s,n] (efc): The GL(2) Fourier series n th term. EisensteinSeriesTerm[z,s,ab] (est): The n th term of the Eisenstein series for GL(2). ElementaryMatrix[n,i,j,c] (elm): Construct a specified elementary matrix. FunctionalEquation[vs, i] (feq): Generate the affine parameter maps. GetCasimirOperator[m,n,"x","y","f"] (gco): The operators for dimension n. GetMatrixElement[a, i, j] (gme): Return a specified element. GlnVersion[] (glv): Print the date of the current version. HeckeEigenvalues[m,n,"a"] (hev): The values of (λ m ) for GL(n). HeckeMultiplicativeSplit[m](hms): Prepare to evaluate a Hecke Fourier coefficient. HeckeCoefficientSum[m, ms, "x"](hcs): The right hand side of the Hecke sum. HeckeOperator[n, z,"f"] (hop): of n th order for square integrable forms on h n . HeckePowerSum[e, es,"B"](hps): Exponents for the Hecke sum at any prime. HermiteFormLower[a] (hfl): The lower left Hermite form. TailingMatrixBlock[a,i,j] (tmb): Extract a tailing matrix block. UpperTriangleToMatrix[u] (utm): Form an upper triangular matrix. VolumeFormDiagonal["a",n] (vfd): The volume form for diagonal matrices. VolumeFormGln["g",n] (vfg): The volume form for GL(n). VolumeFormHn["x","y",n] (vfh): Volume form for the upper half plane. VolumeFormUnimodular["x",n) (vfu): The volume form for the unimodular group. VolumeBall[r,n] (vbl): The volume of a ball in n-dimensions. VolumeHn[n] (vhn): The volume of the generalized upper half plane.