There is an interesting essay:

http://www.claymath.org/cw/arthur/pdf/52.pdf

I would like to raise a question: what are the analogies of:

1) the determiniant

2) the characteristic polynomial

3) the diagonal matrix

And, the final question:

What is the connection between "Geometric Objects" and "Spectral Objects", in a categorical sense?

I am seeking the answers to them, [if they have not been answered].

The title of this essay by James Arthur looks interesting on first sight --- "Harmonic Analysis and Group Representations", but its main topic is the Plancherel formula by Harish-Chandra, which is a very complex result even in the highly sophiscated field of representation theory. The author's choice of notations is not very friendly. I think most of the essay's contents are too abstract to be of general interest, but the "Table of Illustrations" on the second page is very appealing.

Here I just ask some possibly naive questions on the meaning of notations in this essay ---

* SO(p,q;R). This is the first time I saw the concept of special orthogonal groups be applied to non-square rectangular (pxq) matrices. I guess this is no longer rotation groups as used in physics? What puzzles me here is that nonsquare (pxq) matrices don't have determinants any more, then what's the meaning of "special" in this context? Also, the concept of "orthogonal matrices" also becomes meaningless for nonsquare matrices, right? So maybe I misunderstood the meaning of p and q in this notation, they don't refer to the dimension of a matrix any more?

* U(p,q;C). This should refer to the group of unitary complex matrices, right? This seems to confirm that p and q have to denote the dimensions of a matrix.

* L^2(G) and L^2(R/Z). The L^2 notation is very rarely used, Arthur seems to be the only one using this notation and he never explained it. In Harish-Chandra's 1954 PNAS paper, the notation was L_2(G) instead. Neither author provides any explanation of this notation.

According to Joseph Rotman, there are two initial approaches to the subject of representation theory: (1) the character theory approach; and (2) the group algebra approach. James Arthur's definition of a representation on the first page specifies V to be a complex vector space, so I guess he is emphasizing the group algebra approach in this essay.

I think the Plancherel formula is a reflection of the group's topology.

The "Table of Illustrations" is very appealing, and that was what I mean in the first post.

non-square rectangular (pxq) matrices.
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no, it means the sign of the eigenvalues, like SO(3,1) in SR.


L^2
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square integrable.