我們研究定義在某些李群上之不可約表現的兩種指標，分別是Frobenius-Schur指標與Real-Quaternionic指標。前者用來判斷一自對偶表現是對稱或斜對稱；而後者是判斷一自共軛表現是實型或四元數型。本篇論文將在(g,K)-模的層面上分別探討在特殊線性群SL(2,R)，以及在其非線性二重覆疊群中這兩種指標之間的關係。

We study two indicators defined on irreducible representations of certain Lie groups:(a) the Frobenius-Schur indicator ε;(b) the Real-Quaternionic indicator δ. The former is used to determine a self-dual representation is symmetric or skew-symmetric; the later is to test a self-conjugate representation is of real type or quaternionic type. In this paper, we study these two indicators in details for SL(2,R) and its nonlinear double cover on the level of (g,K)-modules.

1 Introduction.......4
2 Representationsof SL(2,R)...... 5
3 Self-dual,Self-conjugate,Hermitian representations of SL(2,R)... 11
3.1 Self-dual representations of SL(2,R) ................ 11
3.2 Self-conjugate representations of SL(2,R) ............. 13
3.3 Hermitian representations of SL(2,R) ............... 16
4 Indicators for SL(2,R)...... 19
5 Representations of SL(2,R)...... 29
6 Self-dual,Self-conjugate,Hermitian representations of SL(2,R)..... 33
6.1 Self-dual representations of SL(2,R) ................ 34
6.2 Self-conjugate representations of SL(2,R) ............. 34
6.3 Hermitian representations of SL(2,R) ............... 35

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