I am reading the book representation theory of semisimple groups. On page 33, the principal series representation $\mathcal{P}^{k,iv}$ is defined as follows. What are motivations of the above formula? Any help will be greatly appreciated!

References request: Ramanujan’s tau function. Let $\Delta(z)=q\prod_{n=1}^{\infty} (1-q^n)^{24}=\sum_{n=1}^{\infty} \tau(n)q^n$, $q=e^{2\pi i z}$. How can one show the following using representation theory? $$ \tau(n)=\sum_S \frac{(a-b)(a-c)(a-d)(a-e)(b-c)(b-d)(b-e)(c-d)(c-e)(d-e)}{1!2!3!4!}, $$ where $S$ is the set of ordered tuples $(a,b,c,d,e)\in \mathbb{Z}^5 $ with $$(a,b,c,d,e) \equiv (1,2,3,4,5) \pmod 5 $$ $$ a+b+c+d+e=0 $$ $$a^2+b^2+c^2+d^2+e^2=10n.$$ Thank you very much.

Assuming that a person has taken standard undergraduate math courses (algebra, analysis, point-set topology), what other things he must know before he can understand the Langlands program and its geometric analogue? What are the good books for learning these topics? Is there any book which can explain the Langlands program to an undergraduate with very […]

I would like to know some reference to learn the theory of automorphic forms. Any (good) book or online lecture notes will be fine. I am particularly interested in the arithmetic point of view (e.g. galois representations associated).

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