Quaternionic gamma functions and their logarithmic derivatives as spectral functions (Q5945797)

Let \(H\) be the algebra of quaternions over \(\mathbb{R}\) with the basis \(\{1,i,j,k\}\), where, as usual, \(k=ij=-ji\), \(i^2=j^2=-1\), and let NEWLINE\[NEWLINEx=x_0+x_1i+x_2j+x_3k,\;|x|=x^2_0+x^2_1+x^2_2+x^2_3NEWLINE\]NEWLINE for \(x\in H\). Let us consider the Hilbert space \(X:=L^2(H,dx)\) with \(dx=4dx_0dx_1dx_2dx_3\), and the group \(G:=H^*\) of invertible elements in \(H\). Inspired by some calculations of \textit{A. Connes} [Sel. Math., New Ser. 5, 29-106 (1999; Zbl 0945.11015)], the author computes the trace \text{Tr} \(A\) of the operator \(A':=FP_\Lambda F^{-1}P_\Lambda U_f\) in \(X\) for any test-function \(f\) of compact support on \(G\), where \(P_\Lambda\) stands for the cut-off projection to functions on \(H\) with support in \(|x|<\Lambda\), \(U_f\) is the bounded operator of multiplicative convolution with \(f\) defined by NEWLINE\[NEWLINE(U_f\varphi)(x)=(2\pi^2)^{-1}\int_Gf(g)\varphi(g^{-1}x)|g|^{-1} dg,NEWLINE\]NEWLINE and \(F\) is the additive Fourier transform operator in \(X\). In the spirit of Tate's thesis [see \textit{J. W. S. Cassels} (ed.) and \textit{A. Fröhlich}, (ed.), Algebraic Number Theory, Academic Press (1967; Zbl 0153.07403), Ch. 15], the author derives te local functional equation and computes the corresponding \(\gamma\)-factors at the infinite place [cf. also \textit{R. Godement} and \textit{H. Jacquet}, Zeta functions of simple algebras, Lecture Notes in Mathematics, 260, Springer-Verlag (1972; Zbl 0244.12011)].