GL_LUMINANCE as a texture format actually does is store one component of data per pixel that is used for the red, green and blue color components. So, a classic greyscale image (It doesn't use it for the alpha component, though, that's why there was also
GL_LUMINANCE_ALPHA for two compnent storage and
GL_INTENSITY that broadcasts it to all components).
When using shaders you don't actually need this distinction anymore, though, since you can do the swizzling yourself in the shader. So you just use a proper typed 1-component internal format, most probably
GL_R8 (i.e. 8-bit normalized fixed point red) in your case, but depending on your exact data type requirements really. And for setting it you would (in that case) use
GL_RED as the format of the data in your RAM storage. So something like:
glTexImage2D(GL_TEXTURE_2D, 0, GL_R8, <width>, <height>, 0, GL_RED, <type>, <data>);
And in the shader you can do with that red color component whatever you want. If you just need a single value per pixel, you use that. If you need a grey color (i.e.
GL_LUMINANCE behaviour), you swizzle it to the RGB components, and if you need it as color and opacity, you swizzle it into all four components. Or whatever else you want to do.
float data = texture(...).r;
vec4 luminance = texture(...).rrra;
vec4 intensity = texture(...).rrrr;
Though, this requires your shader to be somewhat aware of what texture you're working with. But if you want to use a more generic shader working with a 4-component color from an arbitrary texture, this might be too much of a hassle. In this case what you can also do is have the texture itself control its own swizzling behaviour. This is an OpenGL 3.3+ feature and can be set with
glTexParameter and for good old
GL_LUMIMANCE behaviour this would be:
glTexParameter(..., GL_TEXTURE_SWIZZLE_R, GL_RED);
glTexParameter(..., GL_TEXTURE_SWIZZLE_G, GL_RED);
glTexParameter(..., GL_TEXTURE_SWIZZLE_B, GL_RED);
glTexParameter(..., GL_TEXTURE_SWIZZLE_A, GL_ONE);
For a broadcast into all components, i.e.
GL_INTENSITY behaviour, you can use the shortcut
glTexParameter(..., GL_TEXTURE_SWIZZLE_RGBA, GL_RED);
So then you can retrieve a normal 4-component color from your texture in the shader and get the 1-component it's storing conveniently broadcast into the RGBA components of the result.