Hello
As I checked the atlas of finite simple groups if $p$ is a prime then according to the notations of the atlas of finite simple groups, $2.L\_2(p)$ has an irreducible character of degree $(p-1)/2$ or $(p+1)/2$.
Why this character exists and is it true for each prime number $p>3$?
Thanks so much for your help.



The irreducible character of $2.L_2(p)$ where p is a prime
OK, so if I am not mistaken the non-split extension of $L\_2(p)$ should simply be $SL\_2(p)$. You are now asking whether $SL\_2(p)$ has a character of degree $(p-1)/2$ or $(p+1)2$ when $p \neq 2$. Indeed this is true. The generic character table of $SL\_2(q)$ is given in Table 5.4 of Bonnafe's book "Representations of $SL\_2(q)$". We see from this table that the degrees of the irreducible characters of $SL\_2(p)$ are contained in the list


$$1,p,\frac{p+1}{2},\frac{p-1}{2},p+1,p-1$$


There is only one character of degree 1, the trivial character, and one character of degree $p$, the Steinberg character. There are then two characters of degree $\frac{p+1}{2}$ and two characters of degree $\frac{p-1}{2}$. These occur as irreducible constituents of the restriction of an irreducible Deligne-Lusztig character of $GL\_2(p)$. When $p=2$ then one obtains the character table of $SL\_2(p)$ simply by restricting the characters of $GL\_2(p)$.


You may also find this useful by Mark Reeder.


<https://www2.bc.edu/~reederma/SL(2,q).pdf>


