If $a$, $b$ are some events, e.g. $a$: it is raining, $b$: you have your umbrella with you, then, you are asking for P(it is raining and you have umbrella | it is raining). You know that it is raining, so, intuitively, no need to question if it is raining or not again, and this probability equals to P(you have umbrella | it is raining), which will mean $p(a,b|a)=p(b|a)$.
In case of random variables, as @Robin points out, if $A$ and $B$ are the RVs responsible for $a$ and $b$ respectively, the expression should be written in this form: $P(A=a, B=b|A=a)$ in order to prevent any confusions. Because, if $a$ and $b$ are just constants, then what you are asking seems like $p(1,2|1)$, which is ambiguous.
If $a$ and $b$ are random variables themselves, then we lack the particular values. The expression should be like $P(a=a',b=b'|a=a'')$. Here, if $a'' \neq a'$, then given that $a$ equals to $a''$, it cannot be equal to $a'$, and the probability will be $0$. If, $a''=a'$, then again, since we know what the value of $a$ is, the probability will be equal to $p(b=b'|a=a')$, which can be summarized with the help of a $\delta$ function if you like to write it in a compact form, as in @Robin's answer. (P.S. Don't confuse $\delta$ with continuous dirac-delta function).