  We initiate a study of the boundary version of the square-lattice $Q$-state
Potts antiferromagnet, with $Q \in [0,4]$ real, motivated by the fact that the
continuum limit of the corresponding bulk model is a non-compact CFT, closely
related with the $SL(2,\mathbb{R})_k/U(1)$ Euclidian black-hole coset model.
While various types of conformal boundary conditions (discrete and continuous
branes) have been formally identified for the the $SL(2,\mathbb{R})_k/U(1)$
coset CFT, we are only able in this work to identify conformal boundary
conditions (CBC) leading to a discrete boundary spectrum. The conformal
boundary conditions (CBC) we find are of two types. The first is free boundary
Potts spins, for which we confirm an old conjecture for the generating
functions of conformal levels, and show them to be related to characters in a
non-linear deformation of the $W_\infty$ algebra. The second type of CBC -
which corresponds to restricting the values of the Potts spins to a subset of
size $Q_1$, or its complement of size $Q-Q_1$, at alternating sites along the
boundary - is new, and turns out to be conformal in the antiferromagnetic case
only. Using algebraic and numerical techniques, we show that the corresponding
spectrum generating functions produce all the characters of discrete
representations for the coset CFT. The normalizability bounds of the associated
discrete states in the coset CFT are found to have a simple interpretation in
terms of boundary phase transitions in the lattice model. For $\sqrt{Q} = 2
\cos \frac{\pi}{k}$, with $k \ge 3$ integer, we show also how our boundary
conditions can be reformulated in terms of a RSOS height model. The spectrum
generating functions are then identified with string functions of the compact
$SU(2)_{k-2} / U(1)$ parafermion theory (with symmetry $Z_{k-2}$). The new alt
conditions are needed to cover all the string functions.
